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Applications of integration: area, volume, arc length

Areas between curves, solids of revolution, average value.

Every application is the same move: slice the quantity into thin pieces whose size you can write, then add them with an integral. Area between curves is ∫(top − bottom); a solid of revolution is ∫π r² dx (discs); arc length is ∫√(1 + f′²). Picture it: the shaded region under the curve is the integral being computed. Think it: this slicing is the Riemann sum, and measure theory is what makes it rigorous.

Àwọn ààtò àwọn ìṣàmúlò-ètò · integrate pi*(sqrt(x))^2 dx from 0 to 4

Calculus · Definite-Integrals

Integrate pi·x from 0 to 4

\int_{0}^{4} \pi x\, dx

Àwọn ìṣàmúlò-ètò

\int_{0}^{4} \pi x\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
\int \pi x\, dx = \pi \int x\, dx
Pull the constant \pi out of the integral.
\int x\, dx = \frac{x^{2}}{2}
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
F(4) - F(0) = \left(8 \pi\right) - \left(0\right)
Fundamental theorem of calculus: plug in the limits.
= 8 \pi \approx 25.133
Simplify.

Fi àwọn àgbèwọlé hàn

8 \pi

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Bii o ṣe le: Applications of integration: area, volume, arc length

First find an antiderivative F, then evaluate F(b) − F(a).
Pull the constant \pi out of the integral.
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 Definite integrals Taylor series Series and sums Maxima and minima The chain rule Implicit differentiation Related rates and optimisation Integration techniques: substitution, parts, partial fractions Infinite series and convergence tests