Even Functions and Definite Integrals

Calculus Level 1

$f(x)$ and $g(x)$ intersect at $(-c,a)$ and $(c,a)$ .

$g(x)$ is an even function.

$\int_{-c}^{c} f(x) dx$ = 18

$\int_{-c}^{0} g(x) dx$ = 7

What is the area between $f(x)$ and $g(x)$ on $-c$ $\leq$ $0$ $\leq$ $c$ ?

1 solution

Zachary Stewart
May 20, 2018

The area between $f(x)$ and $g(x)$ is $\int_{-c}^{c} [f(x)-g(x)] dx$ .

$g(x)$ is an even function, so $\int_{-c}^{c} g(x) dx$ = $2\cdot\int_{-c}^{0} g(x) dx$ .

Thus, $\int_{-c}^{c} [f(x)-g(x)] dx$

$=$ $\int_{-c}^{c} f(x) dx$ $-$ $\int_{-c}^{c} g(x) dx$

$=$ $18-$ $2\cdot\int_{-c}^{0} g(x) dx$

$=$ $18-$ $2\cdot7$

$=$ $18- 14$

$=$ $4$