Applications of Integration flashcards

Question 1

State the formula for area between two curves $y=f(x)$ and $y=g(x)$ on $[a,b]$ (with $f\ge g$).

Accepted Answer

$$A=\int_a^b \big(f(x)-g(x)\big)\,dx$$
$f$ is the top function, $g$ is the bottom function, and $A\ge0$ is geometric area.

Question 2

State the washer method formula for volume about the $x$-axis.

Accepted Answer

$$V=\pi\int_a^b \Big((R(x))^2-(r(x))^2\Big)\,dx$$
$R(x)$ is the outer radius and $r(x)$ is the inner radius (distance to the “hole”).

Question 3

State the arc length formula for $y=f(x)$ on $[a,b]$.

Accepted Answer

$$L=\int_a^b \sqrt{1+\big(f'(x)\big)^2}\,dx$$
Requires $f'$ to exist (and be integrable) on $[a,b]$.

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