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Definite integrals

Area under a curve and the fundamental theorem of calculus.

A definite integral is the signed area between the curve and the axis on an interval. Find an antiderivative F, then compute F(b) − F(a): the fundamental theorem of calculus. The shaded region in the graph is the area being computed.

Àwọn ààtò àwọn ìṣàmúlò-ètò · integrate x^2 dx from 0 to 1

Calculus · Definite-Integrals

Integrate x^2 from 0 to 1

\int_{0}^{1} x^{2}\, dx

Àwọn ìṣàmúlò-ètò

\int_{0}^{1} x^{2}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
\int x^{2}\, dx = \frac{x^{3}}{3}
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
F(1) - F(0) = \left(\frac{1}{3}\right) - \left(0\right)
Fundamental theorem of calculus: plug in the limits.
= \frac{1}{3}
Simplify.

Fi àwọn àgbèwọlé hàn

\frac{1}{3}

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Bii o ṣe le: Definite integrals

First find an antiderivative F, then evaluate F(b) − F(a).
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
Fundamental theorem of calculus: plug in the limits.
Simplify.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Calculus

Limits Derivatives Integrals Taylor series Series and sums Maxima and minima The chain rule Implicit differentiation Related rates and optimisation Integration techniques: substitution, parts, partial fractions Applications of integration: area, volume, arc length Infinite series and convergence tests