Applications of Integrals

Values and distance

Average value

f_{avg} = \frac{1}{b - a} \int_{a}^{b} f (x) d x

Total distance traveled

\int_{t_{1}}^{t_{2}} | v (t) | d t

Area between curves

A = \int_{a}^{b} [f (x) - g (x)] d x

f(x) ≥ g(x)

Volume by disks or washers

V = π \int_{a}^{b} ([R (x)]^{2} - [r (x)]^{2}) d x

Volume by shells

V = 2 π \int_{a}^{b} x f (x) d x

Arc length

L = \int_{a}^{b} \sqrt{1 + [f^{'} (x)]^{2}} d x

Arc length, parametric

L = \int_{t_{1}}^{t_{2}} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t

Polar area

A = \frac{1}{2} \int_{α}^{β} [r (θ)]^{2} d θ

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