Volumes and arc length

Japanese school year: Math III

What you learn

In this section, you will learn how to calculate solid volumes, volumes of revolution, and arc lengths using definite integrals. These techniques are essential for geometric measurements and physical modeling. A solid grasp of basic definite integration and derivatives is recommended before starting this topic.

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Key points

This is the fundamental formula for finding the volume $V$ of a solid. It integrates the cross-sectional area $S(x)$ , sliced perpendicular to the $x$ -axis, across the interval from $x=a$ to $x=b$ .

V = \int_{a}^{b} S(x) \, dx

This formula calculates the volume $V$ of a solid formed by rotating the region under $y=f(x)$ around the $x$ -axis. Each cross section is a circular disk with radius $|f(x)|$ and area $\pi \{f(x)\}^2$ , integrated from $x=a$ to $x=b$ .

V = \pi \int_{a}^{b} \{f(x)\}^2 \, dx

This formula calculates the arc length $L$ of a curve $y=f(x)$ from $x=a$ to $x=b$ . Based on integrating tiny hypotenuses via the Pythagorean theorem, it uses the derivative $f'(x)$ , which represents the curve's slope.

L = \int_{a}^{b} \sqrt{1 + \{f'(x)\}^2} \, dx

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Practice 1Coming soon