Calculus - Applications of Integrals

Applications of Integrals

Area Between Curves

$\text{Area} = \int_a^b [f(x) - g(x)]dx$

where f(x) ≥ g(x) on [a,b].

Volumes

Disk Method: $V = \int_a^b \pi[f(x)]^2 dx$

Washer Method: $V = \int_a^b \pi([f(x)]^2 - [g(x)]^2)dx$

Shell Method: $V = \int_a^b 2\pi xh(x)dx$ or $V = \int_c^d 2\pi yw(y)dy$

Arc Length

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

For parametric x(t), y(t): $L = \int_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2}dt$

Surface Area of Revolution

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

Work

$W = \int_a^b F(x)dx$

Pressure and Force

$F = \int_a^b \rho h(x)w(x)dx$