Applications of Integration

Area Between Curves

$A = \int_a^b |f(x) - g(x)|\,dx$ or for curves described as functions of $y$: $A = \int_c^d |f(y) - g(y)|\,dy$

Volume: Disk Method

Rotating $y = f(x)$ around the $x$-axis: $V = \pi\int_a^b [f(x)]^2\,dx$

Volume: Washer Method

Rotating the region between $f(x)$ and $g(x)$ ($f > g \geq 0$): $V = \pi\int_a^b \left([f(x)]^2 - [g(x)]^2\right)dx$

Volume: Shell Method

Rotating around the $y$-axis: $V = 2\pi\int_a^b x \cdot f(x)\,dx$

Volume by Cross-Sections

If cross-sections perpendicular to the $x$-axis have area $A(x)$: $V = \int_a^b A(x)\,dx$

Common cross-sections: squares, semicircles, equilateral triangles.

Arc Length

$L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$

Problem: Find the volume when $y = \sqrt{x}$ from $x = 0$ to $x = 4$ is rotated about the $x$-axis.

Solution: Disk method: $V = \pi\int_0^4 (\sqrt{x})^2\,dx = \pi\int_0^4 x\,dx = \pi\left[\frac{x^2}{2}\right]_0^4 = \pi(8) = 8\pi$

• Area: $\int |f - g|$.
• Volume of revolution: disk ($\pi r^2$), washer ($\pi(R^2 - r^2)$), shell ($2\pi rh$).
• Volume by cross-sections: $\int A(x)\,dx$.
• Arc length: $\int\sqrt{1 + (f')^2}\,dx$.