Question

Suppose
`f` is an even function and
`\int\limits^7_(-7) {f(x)} \, dx =18`.

a. Evaluate
`\int\limits^7_0 {f(x)} \, dx`
b. Evaluate
`\int\limits^7_(-7) {xf(x)} \, dx`

Answer 1

a. Because
`f` is even,

`\displaystyle \int_(-7)^7 f(x) \, dx = 2 \int_0^7 f(x) \, dx = 18 \\\\ ~~~~ \implies \int_0^7 f(x) \, dx = \frac{18}2 = \boxed{9}`

b. The product of an even function
`(f)` by an odd function
`(x)` is an odd function, whose integral over a symmetric interval is zero.

`\displaystyle \int_(-7)^7 xf(x) \, dx = \boxed{0}`