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Approximating Areas

Use sigma (summation) notation to calculate sums and powers of integers.

Sigma (Summation) Notation

As mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. This process often requires adding up long strings of numbers. To make it easier to write down these lengthy sums, we look at some new notation here, called sigma notation (also known as summation notation). The Greek capital letter \(\Sigma ,\) sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write

\[1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20.\]

We could probably skip writing a couple of terms and write

\[1+2+3+4+\text{\cdots }+19+20,\]

which is better, but still cumbersome. With sigma notation, we write this sum as

\[\sum _{i=1}^{20}i,\]

which is much more compact.

Typically, sigma notation is presented in the form

\[\sum _{i=1}^{n}{a}_{i}\]

where \({a}_{i}\) describes the terms to be added, and the i is called the index. Each term is evaluated, then we sum all the values, beginning with the value when \(i=1\) and ending with the value when \(i=n.\) For example, an expression like \(\sum _{i=2}^{7}{s}_{i}\) is interpreted as \({s}_{2}+{s}_{3}+{s}_{4}+{s}_{5}+{s}_{6}+{s}_{7}.\) Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a dummy variable. We can use any letter we like for the index. Typically, mathematicians use i, j, k, m, and n for indices.

Let’s try a couple of examples of using sigma notation.

Example

Try it.

  1. Write in sigma notation and evaluate the sum of terms \({3}^{i}\) for \(i=1,2,3,4,5.\)
  2. Write the sum in sigma notation:
    \[1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}.\]
Solution
  1. Write
    \[\begin{array}{ll}\sum _{i=1}^{5}{3}^{i} & =3+{3}^{2}+{3}^{3}+{3}^{4}+{3}^{5} \\ & =363.\end{array}\]
  2. The denominator of each term is a perfect square. Using sigma notation, this sum can be written as \(\sum _{i=1}^{5}\frac{1}{{i}^{2}}.\)

Condensed — the full section is in OpenStax Calculus Volume 2.

Approximating Area

Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let \(f(x)\) be a continuous, nonnegative function defined on the closed interval \([a,b].\) We want to approximate the area A bounded by \(f(x)\) above, the x-axis below, the line \(x=a\) on the left, and the line \(x=b\) on the right ().

How do we approximate the area under this curve? The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area. We begin by dividing the interval \([a,b]\) into n subintervals of equal width, \(\frac{b-a}{n}.\) We do this by selecting equally spaced points \({x}_{0},{x}_{1},{x}_{2}\text{,\ldots ,}\ {x}_{n}\) with \({x}_{0}=a,{x}_{n}=b,\) and

\[{x}_{i}-{x}_{i-1}=\frac{b-a}{n}\]

for \(i=1,2,3\text{,\ldots ,}\ n.\)

We denote the width of each subinterval with the notation Δx, so \(\text{\Delta }x=\frac{b-a}{n}\) and

\[{x}_{i}={x}_{0}+i\text{\Delta }x\]

for \(i=1,2,3\text{,\ldots ,}\ n.\) This notion of dividing an interval \([a,b]\) into subintervals by selecting points from within the interval is used quite often in approximating the area under a curve, so let’s define some relevant terminology.

We can use this regular partition as the basis of a method for estimating the area under the curve. We next examine two methods: the left-endpoint approximation and the right-endpoint approximation.

The second method for approximating area under a curve is the right-endpoint approximation. It is almost the same as the left-endpoint approximation, but now the heights of the rectangles are determined by the function values at the right of each subinterval.

Condensed — the full section is in OpenStax Calculus Volume 2.

Forming Riemann Sums

So far we have been using rectangles to approximate the area under a curve. The heights of these rectangles have been determined by evaluating the function at either the right or left endpoints of the subinterval \([{x}_{i-1},{x}_{i}].\) In reality, there is no reason to restrict evaluation of the function to one of these two points only. We could evaluate the function at any point \({x}_{i}^{*}\) in the subinterval \([{x}_{i-1},{x}_{i}],\) and use \(f({x}_{i}^{*})\) as the height of our rectangle. This gives us an estimate for the area of the form

\[A\approx \sum _{i=1}^{n}f({x}_{i}^{*})\text{\Delta }x.\]

A sum of this form is called a Riemann sum, named for the 19th-century mathematician Bernhard Riemann, who developed the idea.

Recall that with the left- and right-endpoint approximations, the estimates seem to get better and better as n get larger and larger. The same thing happens with Riemann sums. Riemann sums give better approximations for larger values of n. We are now ready to define the area under a curve in terms of Riemann sums.

Some subtleties here are worth discussing. First, note that taking the limit of a sum is a little different from taking the limit of a function \(f(x)\) as x goes to infinity. Limits of sums are discussed in detail in the chapter on Sequences and Series; however, for now we can assume that the computational techniques we used to compute limits of functions can also be used to calculate limits of sums.

Second, we must consider what to do if the expression converges to different limits for different choices of \(\{{x}_{i}^{*}\}.\) Fortunately, this does not happen. Although the proof is beyond the scope of this text, it can be shown that if \(f(x)\) is continuous on the closed interval \([a,b],\) then \(\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}f({x}_{i}^{*})\text{\Delta }x\) exists and is unique (in other words, it does not depend on the choice of \(\{{x}_{i}^{*}\}\text{).}\)

Condensed — the full section is in OpenStax Calculus Volume 2.

Key Concepts

  • The use of sigma (summation) notation of the form \(\sum _{i=1}^{n}{a}_{i}\) is useful for expressing long sums of values in compact form.
  • For a continuous function defined over an interval \([a,b],\) the process of dividing the interval into n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
  • The width of each rectangle is \(\text{\Delta }x=\frac{b-a}{n}.\)
  • Riemann sums are expressions of the form \(\sum _{i=1}^{n}f({x}_{i}^{*})\text{\Delta }x,\) and can be used to estimate the area under the curve \(y=f(x).\) Left- and right-endpoint approximations are special kinds of Riemann sums where the values of \(\{{x}_{i}^{*}\}\) are chosen to be the left or right endpoints of the subintervals, respectively.
  • Riemann sums allow for much flexibility in choosing the set of points \(\{{x}_{i}^{*}\}\) at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.

Key Equations

Properties of Sigma Notation\(\sum _{i=1}^{n}c=nc\)
\(\sum _{i=1}^{n}c{a}_{i}=c\sum _{i=1}^{n}{a}_{i}\)
\(\sum _{i=1}^{n}({a}_{i}+{b}_{i})=\sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}\)
\(\sum _{i=1}^{n}({a}_{i}-{b}_{i})=\sum _{i=1}^{n}{a}_{i}-\sum _{i=1}^{n}{b}_{i}\)
\(\sum _{i=1}^{n}{a}_{i}=\sum _{i=1}^{m}{a}_{i}+\sum _{i=m+1}^{n}{a}_{i}\)
Sums and Powers of Integers\(\sum _{i=1}^{n}i=1+2+\text{\cdots }+n=\frac{n(n+1)}{2}\)
\(\sum _{i=1}^{n}{i}^{2}={1}^{2}+{2}^{2}+\text{\cdots }+{n}^{2}=\frac{n(n+1)(2n+1)}{6}\)
\(\sum _{i=0}^{n}{i}^{3}={1}^{3}+{2}^{3}+\text{\cdots }+{n}^{3}=\frac{{n}^{2}{(n+1)}^{2}}{4}\)
Left-Endpoint Approximation\(A\approx {L}_{n}=f({x}_{0})\text{\Delta }x+f({x}_{1})\text{\Delta }x+\text{\cdots }+f({x}_{n-1})\text{\Delta }x=\sum _{i=1}^{n}f({x}_{i-1})\text{\Delta }x\)
Right-Endpoint Approximation\(A\approx {R}_{n}=f({x}_{1})\text{\Delta }x+f({x}_{2})\text{\Delta }x+\text{\cdots }+f({x}_{n})\text{\Delta }x=\sum _{i=1}^{n}f({x}_{i})\text{\Delta }x\)

Approximating Areas

In the following exercises, use the rules for sums of powers of integers to compute the sums.

Suppose that \(\sum _{i=1}^{100}{a}_{i}=15\) and \(\sum _{i=1}^{100}{b}_{i}=-12.\) In the following exercises, compute the sums.

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

Let \({L}_{n}\) denote the left-endpoint sum using n subintervals and let \({R}_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.

Express the following endpoint sums in sigma notation but do not evaluate them.

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L8.

Condensed — the full section is in OpenStax Calculus Volume 2.

Sigma (Summation) Notation

As mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. This process often requires adding up long strings of numbers. To make it easier to write down these lengthy sums, we look at some new notation here, called sigma notation (also known as summation notation). The Greek capital letter \(\Sigma ,\) sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write

\[1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20.\]

We could probably skip writing a couple of terms and write

\[1+2+3+4+\text{\cdots }+19+20,\]

which is better, but still cumbersome. With sigma notation, we write this sum as

\[\sum _{i=1}^{20}i,\]

which is much more compact.

Typically, sigma notation is presented in the form

\[\sum _{i=1}^{n}{a}_{i}\]

where \({a}_{i}\) describes the terms to be added, and the i is called the index. Each term is evaluated, then we sum all the values, beginning with the value when \(i=1\) and ending with the value when \(i=n.\) For example, an expression like \(\sum _{i=2}^{7}{s}_{i}\) is interpreted as \({s}_{2}+{s}_{3}+{s}_{4}+{s}_{5}+{s}_{6}+{s}_{7}.\) Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a dummy variable. We can use any letter we like for the index. Typically, mathematicians use i, j, k, m, and n for indices.

Let’s try a couple of examples of using sigma notation.

Example

Try it.

  1. Write in sigma notation and evaluate the sum of terms \({3}^{i}\) for \(i=1,2,3,4,5.\)
  2. Write the sum in sigma notation:
    \[1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}.\]
Solution
  1. Write
    \[\begin{array}{ll}\sum _{i=1}^{5}{3}^{i} & =3+{3}^{2}+{3}^{3}+{3}^{4}+{3}^{5} \\ & =363.\end{array}\]
  2. The denominator of each term is a perfect square. Using sigma notation, this sum can be written as \(\sum _{i=1}^{5}\frac{1}{{i}^{2}}.\)

Condensed — the full section is in OpenStax Calculus Volume 1.

Approximating Area

Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let \(f(x)\) be a continuous, nonnegative function defined on the closed interval \([a,b].\) We want to approximate the area A bounded by \(f(x)\) above, the x-axis below, the line \(x=a\) on the left, and the line \(x=b\) on the right ().

How do we approximate the area under this curve? The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area. We begin by dividing the interval \([a,b]\) into n subintervals of equal width, \(\frac{b-a}{n}.\) We do this by selecting equally spaced points \({x}_{0},{x}_{1},{x}_{2}\text{,\ldots ,}\ {x}_{n}\) with \({x}_{0}=a,{x}_{n}=b,\) and

\[{x}_{i}-{x}_{i-1}=\frac{b-a}{n}\]

for \(i=1,2,3\text{,\ldots ,}\ n.\)

We denote the width of each subinterval with the notation Δx, so \(\text{\Delta }x=\frac{b-a}{n}\) and

\[{x}_{i}={x}_{0}+i\text{\Delta }x\]

for \(i=1,2,3\text{,\ldots ,}\ n.\) This notion of dividing an interval \([a,b]\) into subintervals by selecting points from within the interval is used quite often in approximating the area under a curve, so let’s define some relevant terminology.

We can use this regular partition as the basis of a method for estimating the area under the curve. We next examine two methods: the left-endpoint approximation and the right-endpoint approximation.

The second method for approximating area under a curve is the right-endpoint approximation. It is almost the same as the left-endpoint approximation, but now the heights of the rectangles are determined by the function values at the right of each subinterval.

Condensed — the full section is in OpenStax Calculus Volume 1.

Forming Riemann Sums

So far we have been using rectangles to approximate the area under a curve. The heights of these rectangles have been determined by evaluating the function at either the right or left endpoints of the subinterval \([{x}_{i-1},{x}_{i}].\) In reality, there is no reason to restrict evaluation of the function to one of these two points only. We could evaluate the function at any point \({x}_{i}^{*}\) in the subinterval \([{x}_{i-1},{x}_{i}],\) and use \(f({x}_{i}^{*})\) as the height of our rectangle. This gives us an estimate for the area of the form

\[A\approx \sum _{i=1}^{n}f({x}_{i}^{*})\text{\Delta }x.\]

A sum of this form is called a Riemann sum, named for the 19th-century mathematician Bernhard Riemann, who developed the idea.

Recall that with the left- and right-endpoint approximations, the estimates seem to get better and better as n get larger and larger. The same thing happens with Riemann sums. Riemann sums give better approximations for larger values of n. We are now ready to define the area under a curve in terms of Riemann sums.

Some subtleties here are worth discussing. First, note that taking the limit of a sum is a little different from taking the limit of a function \(f(x)\) as x goes to infinity. Limits of sums are discussed in detail in the chapter on Sequences and Series; however, for now we can assume that the computational techniques we used to compute limits of functions can also be used to calculate limits of sums.

Second, we must consider what to do if the expression converges to different limits for different choices of \(\{{x}_{i}^{*}\}.\) Fortunately, this does not happen. Although the proof is beyond the scope of this text, it can be shown that if \(f(x)\) is continuous on the closed interval \([a,b],\) then \(\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}f({x}_{i}^{*})\text{\Delta }x\) exists and is unique (in other words, it does not depend on the choice of \(\{{x}_{i}^{*}\}\text{).}\)

Condensed — the full section is in OpenStax Calculus Volume 1.

Key Concepts

  • The use of sigma (summation) notation of the form \(\sum _{i=1}^{n}{a}_{i}\) is useful for expressing long sums of values in compact form.
  • For a continuous function defined over an interval \([a,b],\) the process of dividing the interval into n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
  • The width of each rectangle is \(\text{\Delta }x=\frac{b-a}{n}.\)
  • Riemann sums are expressions of the form \(\sum _{i=1}^{n}f({x}_{i}^{*})\text{\Delta }x,\) and can be used to estimate the area under the curve \(y=f(x).\) Left- and right-endpoint approximations are special kinds of Riemann sums where the values of \(\{{x}_{i}^{*}\}\) are chosen to be the left or right endpoints of the subintervals, respectively.
  • Riemann sums allow for much flexibility in choosing the set of points \(\{{x}_{i}^{*}\}\) at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.

Key Equations

Properties of Sigma Notation\(\sum _{i=1}^{n}c=nc\)
\(\sum _{i=1}^{n}c{a}_{i}=c\sum _{i=1}^{n}{a}_{i}\)
\(\sum _{i=1}^{n}({a}_{i}+{b}_{i})=\sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}\)
\(\sum _{i=1}^{n}({a}_{i}-{b}_{i})=\sum _{i=1}^{n}{a}_{i}-\sum _{i=1}^{n}{b}_{i}\)
\(\sum _{i=1}^{n}{a}_{i}=\sum _{i=1}^{m}{a}_{i}+\sum _{i=m+1}^{n}{a}_{i}\)
Sums and Powers of Integers\(\sum _{i=1}^{n}i=1+2+\text{\cdots }+n=\frac{n(n+1)}{2}\)
\(\sum _{i=1}^{n}{i}^{2}={1}^{2}+{2}^{2}+\text{\cdots }+{n}^{2}=\frac{n(n+1)(2n+1)}{6}\)
\(\sum _{i=0}^{n}{i}^{3}={1}^{3}+{2}^{3}+\text{\cdots }+{n}^{3}=\frac{{n}^{2}{(n+1)}^{2}}{4}\)
Left-Endpoint Approximation\(A\approx {L}_{n}=f({x}_{0})\text{\Delta }x+f({x}_{1})\text{\Delta }x+\text{\cdots }+f({x}_{n-1})\text{\Delta }x=\sum _{i=1}^{n}f({x}_{i-1})\text{\Delta }x\)
Right-Endpoint Approximation\(A\approx {R}_{n}=f({x}_{1})\text{\Delta }x+f({x}_{2})\text{\Delta }x+\text{\cdots }+f({x}_{n})\text{\Delta }x=\sum _{i=1}^{n}f({x}_{i})\text{\Delta }x\)

Approximating Areas

In the following exercises, use the rules for sums of powers of integers to compute the sums.

Suppose that \(\sum _{i=1}^{100}{a}_{i}=15\) and \(\sum _{i=1}^{100}{b}_{i}=-12.\) In the following exercises, compute the sums.

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

Let \({L}_{n}\) denote the left-endpoint sum using n subintervals and let \({R}_{n}\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.

Express the following endpoint sums in sigma notation but do not evaluate them.

In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. If the two agree, say "neither."

In the following exercises, estimate the areas under the curves by computing the left Riemann sums, L8.

Condensed — the full section is in OpenStax Calculus Volume 1.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

    1. Write in sigma notation and evaluate the sum of terms \({3}^{i}\) for \(i=1,2,3,4,5.\)
    2. Write the sum in sigma notation:
      \[1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}.\]
    Ҷавоби ҷавобро нишон диҳед
    1. Write
      \[\begin{array}{ll}\sum _{i=1}^{5}{3}^{i} & =3+{3}^{2}+{3}^{3}+{3}^{4}+{3}^{5} \\ & =363.\end{array}\]
    2. The denominator of each term is a perfect square. Using sigma notation, this sum can be written as \(\sum _{i=1}^{5}\frac{1}{{i}^{2}}.\)

  2. Write in sigma notation and evaluate the sum of terms 2i for \(i=3,4,5,6.\)

    Ҷавоби ҷавобро нишон диҳед

    \(\sum _{i=3}^{6}{2}^{i}={2}^{3}+{2}^{4}+{2}^{5}+{2}^{6}=120\)

  3. Write using sigma notation and evaluate:

    1. The sum of the terms \({(i-3)}^{2}\) for \(i=1,2\text{,\ldots ,}\ 200.\)
    2. The sum of the terms \(({i}^{3}-{i}^{2})\) for \(i=1,2,3,4,5,6.\)
    Ҷавоби ҷавобро нишон диҳед
    1. Multiplying out \({(i-3)}^{2},\) we can break the expression into three terms.
      \[\begin{array}{ll}\sum _{i=1}^{200}{(i-3)}^{2} & =\sum _{i=1}^{200}({i}^{2}-6i+9) \\ \\ \\ & =\sum _{i=1}^{200}{i}^{2}-\sum _{i=1}^{200}6i+\sum _{i=1}^{200}9 \\ & =\sum _{i=1}^{200}{i}^{2}-6\sum _{i=1}^{200}i+\sum _{i=1}^{200}9 \\ & =\frac{200(200+1)(400+1)}{6}-6[\frac{200(200+1)}{2}]+9(200) \\ & =2,686,700-120,600+1800 \\ & =2,567,900\end{array}\]
    2. Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.
      \[\begin{array}{ll}\sum _{i=1}^{6}({i}^{3}-{i}^{2}) & =\sum _{i=1}^{6}{i}^{3}-\sum _{i=1}^{6}{i}^{2} \\ \\ \\ \\ & =\frac{{6}^{2}{(6+1)}^{2}}{4}-\frac{6(6+1)(2(6)+1)}{6} \\ & =\frac{1764}{4}-\frac{546}{6} \\ & =350\end{array}\]

  4. Find the sum of the values of \(4+3i\) for \(i=1,2\text{,\ldots ,}\ 100.\)

    Ҷавоби ҷавобро нишон диҳед

    15,550

  5. Find the sum of the values of \(f(x)={x}^{3}\) over the integers \(1,2,3\text{,\ldots ,}\ 10.\)

    Ҷавоби ҷавобро нишон диҳед

    Using the formula, we have

    \[\begin{array}{ll}\sum _{i=1}^{10}{i}^{3} & =\frac{{(10)}^{2}{(10+1)}^{2}}{4} \\ \\ & =\frac{100(121)}{4} \\ & =3025.\end{array}\]

  6. Evaluate the sum indicated by the notation \(\sum _{k=1}^{20}(2k+1).\)

    Ҷавоби ҷавобро нишон диҳед

    440

  7. Use both left-endpoint and right-endpoint approximations to approximate the area under the curve of \(f(x)={x}^{2}\) on the interval \([0,2];\) use \(n=4.\)

    Ҷавоби ҷавобро нишон диҳед

    First, divide the interval \([0,2]\) into n equal subintervals. Using \(n=4,\text{\Delta }x=\frac{(2-0)}{4}=0.5.\) This is the width of each rectangle. The intervals \([0,0.5],[0.5,1],[1,1.5],[1.5,2]\) are shown in . Using a left-endpoint approximation, the heights are \(f(0)=0,f(0.5)=0.25,f(1)=1,f(1.5)=2.25.\) Then,

    \[\begin{array}{ll}{L}_{4} & =f({x}_{0})\text{\Delta }x+f({x}_{1})\text{\Delta }x+f({x}_{2})\text{\Delta }x+f({x}_{3})\text{\Delta }x \\ & =0(0.5)+0.25(0.5)+1(0.5)+2.25(0.5) \\ & =1.75.\end{array}\]

    The right-endpoint approximation is shown in . The intervals are the same, \(\text{\Delta }x=0.5,\) but now use the right endpoint to calculate the height of the rectangles. We have

    \[\begin{array}{ll}{R}_{4} & =f({x}_{1})\text{\Delta }x+f({x}_{2})\text{\Delta }x+f({x}_{3})\text{\Delta }x+f({x}_{4})\text{\Delta }x \\ & =0.25(0.5)+1(0.5)+2.25(0.5)+4(0.5) \\ & =3.75.\end{array}\]

    The left-endpoint approximation is 1.75; the right-endpoint approximation is 3.75.

  8. Sketch left-endpoint and right-endpoint approximations for \(f(x)=\frac{1}{x}\) on \([1,2];\) use \(n=4.\) Approximate the area using both methods.

    Ҷавоби ҷавобро нишон диҳед

    The left-endpoint approximation is 0.7595. The right-endpoint approximation is 0.6345.

  9. Find a lower sum for \(f(x)=10-{x}^{2}\) on \([1,2];\) let \(n=4\) subintervals.

    Ҷавоби ҷавобро нишон диҳед

    With \(n=4\) over the interval \([1,2],\text{\Delta }x=\frac{1}{4}.\) We can list the intervals as \([1,1.25],[1.25,1.5],[1.5,1.75],[1.75,2].\) Because the function is decreasing over the interval \([1,2],\) shows that a lower sum is obtained by using the right endpoints.

    The Riemann sum is

    \[\begin{array}{ll}\sum _{k=1}^{4}(10-{{x}_{k}}^{2})(0.25) & =0.25[10-{(1.25)}^{2}+10-{(1.5)}^{2}+10-{(1.75)}^{2}+10-{(2)}^{2}] \\ & =0.25[8.4375+7.75+6.9375+6] \\ & =7.28.\end{array}\]

    The area of 7.28 is a lower sum and an underestimate.

    1. Find an upper sum for \(f(x)=10-{x}^{2}\) on \([1,2];\) let \(n=4.\)
    2. Sketch the approximation.
    Ҷавоби ҷавобро нишон диҳед
    1. \(\text{Upper sum}=8.0313.\)


  11. Find a lower sum for \(f(x)=\text{sin}\ x\) over the interval \([a,b]=[0,\frac{\pi }{2}];\) let \(n=6.\)

    Ҷавоби ҷавобро нишон диҳед

    Let’s first look at the graph in to get a better idea of the area of interest.

    The intervals are \([0,\frac{\pi }{12}],[\frac{\pi }{12},\frac{\pi }{6}],[\frac{\pi }{6},\frac{\pi }{4}],[\frac{\pi }{4},\frac{\pi }{3}],[\frac{\pi }{3},\frac{5\pi }{12}],\) and \([\frac{5\pi }{12},\frac{\pi }{2}].\) Note that \(f(x)=\text{sin}\ x\) is increasing on the interval \([0,\frac{\pi }{2}],\) so a left-endpoint approximation gives us the lower sum. A left-endpoint approximation is the Riemann sum \(\sum _{i=0}^{5}\text{sin}\ {x}_{i}(\frac{\pi }{12}).\) We have

    \[\begin{array}{ll}A & \approx \text{sin}(0)(\frac{\pi }{12})+\text{sin}(\frac{\pi }{12})(\frac{\pi }{12})+\text{sin}(\frac{\pi }{6})(\frac{\pi }{12})+\text{sin}(\frac{\pi }{4})(\frac{\pi }{12})+\text{sin}(\frac{\pi }{3})(\frac{\pi }{12})+\text{sin}(\frac{5\pi }{12})(\frac{\pi }{12}) \\ & =0.863.\end{array}\]

  12. Using the function \(f(x)=\text{sin}\ x\) over the interval \([0,\frac{\pi }{2}],\) find an upper sum; let \(n=6.\)

    Ҷавоби ҷавобро нишон диҳед

    \(A\approx 1.125\)

  13. State whether the given sums are equal or unequal.

    1. \(\sum _{i=1}^{10}i\) and \(\sum _{k=1}^{10}k\)
    2. \(\sum _{i=1}^{10}i\) and \(\sum _{i=6}^{15}(i-5)\)
    3. \(\sum _{i=1}^{10}i(i-1)\) and \(\sum _{j=0}^{9}(j+1)j\)
    4. \(\sum _{i=1}^{10}i(i-1)\) and \(\sum _{k=1}^{10}({k}^{2}-k)\)
    Ҷавоби ҷавобро нишон диҳед

    a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting \(j=i-1.\) d. They are equal; the first sum factors the terms of the second.

  14. \(\sum _{i=5}^{10}i\)

  15. \(\sum _{i=5}^{10}{i}^{2}\)

    Ҷавоби ҷавобро нишон диҳед

    \(385-30=355\)

  16. \(\sum _{i=1}^{100}({a}_{i}+{b}_{i})\)

  17. \(\sum _{i=1}^{100}({a}_{i}-{b}_{i})\)

    Ҷавоби ҷавобро нишон диҳед

    \(15-(-12)=27\)

  18. \(\sum _{i=1}^{100}(3{a}_{i}-4{b}_{i})\)

  19. \(\sum _{i=1}^{100}(5{a}_{i}+4{b}_{i})\)

    Ҷавоби ҷавобро нишон диҳед

    \(5(15)+4(-12)=27\)

  20. \(\sum _{k=1}^{20}100({k}^{2}-5k+1)\)

  21. \(\sum _{j=1}^{50}({j}^{2}-2j)\)

    Ҷавоби ҷавобро нишон диҳед

    \(\sum _{j=1}^{50}{j}^{2}-2\sum _{j=1}^{50}j=\frac{(50)(51)(101)}{6}-\frac{2(50)(51)}{2}=40,\text{}375\)

  22. \(\sum _{j=11}^{20}({j}^{2}-10j)\)

  23. \(\sum _{k=1}^{25}[{(2k)}^{2}-100k]\)

    Ҷавоби ҷавобро нишон диҳед

    \(4\sum _{k=1}^{25}{k}^{2}-100\sum _{k=1}^{25}k=\frac{4(25)(26)(51)}{6}-50(25)(26)=-10,\text{}400\)

  24. L4 for \(f(x)=\frac{1}{x-1}\) on \([2,3]\)

  25. R4 for \(g(x)=\text{cos}(\pi x)\) on \([0,1]\)

    Ҷавоби ҷавобро нишон диҳед

    \({R}_{4}=-0.25\)

  26. L6 for \(f(x)=\frac{1}{x(x-1)}\) on \([2,5]\)

  27. R6 for \(f(x)=\frac{1}{x(x-1)}\) on \([2,5]\)

    Ҷавоби ҷавобро нишон диҳед

    \({R}_{6}=0.372\)

  28. R4 for \(\frac{1}{{x}^{2}+1}\) on \([-2,2]\)

  29. L4 for \(\frac{1}{{x}^{2}+1}\) on \([-2,2]\)

    Ҷавоби ҷавобро нишон диҳед

    \({L}_{4}=2.20\)

  30. R8 for \({x}^{2}-2x+1\) on \([0,2]\)

  31. L8 for \({x}^{2}-2x+1\) on \([0,2]\)

    Ҷавоби ҷавобро нишон диҳед

    \({L}_{8}=0.6875\)

  32. Compute the left and right Riemann sums—L4 and R4, respectively—for \(f(x)=(2-|x|)\) on \([-2,2].\) Compute their average value and compare it with the area under the graph of f.

  33. Compute the left and right Riemann sums—L6 and R6, respectively—for \(f(x)=(3-|3-x|)\) on \([0,6].\) Compute their average value and compare it with the area under the graph of f.

    Ҷавоби ҷавобро нишон диҳед

    \({L}_{6}=9.000={R}_{6}.\) The graph of f is a triangle with area 9.

  34. Compute the left and right Riemann sums—L4 and R4, respectively—for \(f(x)=\sqrt{4-{x}^{2}}\) on \([-2,2]\) and compare their values.

  35. Compute the left and right Riemann sums—L6 and R6, respectively—for \(f(x)=\sqrt{9-{(x-3)}^{2}}\) on \([0,6]\) and compare their values.

    Ҷавоби ҷавобро нишон диҳед

    \({L}_{6}=13.12899={R}_{6}.\) They are equal.

  36. L30 for \(f(x)={x}^{2}\) on \([1,2]\)

  37. L10 for \(f(x)=\sqrt{4-{x}^{2}}\) on \([-2,2]\)

    Ҷавоби ҷавобро нишон диҳед

    \({L}_{10}=\frac{4}{10}\sum _{i=1}^{10}\sqrt{4-(-2+4\frac{(i-1)}{10}{)}^{2}}\)

  38. R20 for \(f(x)=\text{sin}\ x\) on \([0,\pi ]\)

  39. R100 for \(\text{ln}\ x\) on \([1,e]\)

    Ҷавоби ҷавобро нишон диҳед

    \({R}_{100}=\frac{e-1}{100}\sum _{i=1}^{100}\text{ln}(1+(e-1)\frac{i}{100})\)

  40. [T] L100 and R100 for \(y={x}^{2}-x+3\) on the interval \([-1,1]\)

Symbols used here

\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
i
imaginary unit
i² = −1.
\approx
approximately equal
Equal to the precision shown, not exactly.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
C,\ C_1,\ C_2
arbitrary constants
Constants of integration fixed by initial conditions.

How to: Approximating Areas

  1. Use sigma (summation) notation to calculate sums and powers of integers.
  2. Use the sum of rectangular areas to approximate the area under a curve.
  3. Use Riemann sums to approximate area.
  4. Write in sigma notation and evaluate the sum of terms
  5. Write the sum in sigma notation:
  6. Write
  7. The denominator of each term is a perfect square. Using sigma notation, this sum can be written as
  8. The sum of

Questions people ask

What is a derivative in one sentence?

The slope of the graph at a point — the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate — the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function — a polynomial times an exponential, log or trig function.

Санҷиши худ

Parts of this page are adapted from OpenStax Calculus Volume 1 (CC BY-NC-SA 4.0), OpenStax Calculus Volume 2 (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

& Иловаи забон Calculus

LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests