Question

How do you evaluate the definite integral `int|x^3+64| dx` from `[-8,0]`?

Answer 1

`896`

Explanation:

`f(x) = x^3+64` changes sign for `x = -4` so

`int_{-8}^0abs(f(x))dx = -int_{-8}^{-4}f(x)dx+int_{-4}^0f(x)dx`

but

`int_a^bf(x)dx = 64 b + b^4/4-(64 a + a^4/4)` then

`int_{-8}^0 abs(f(x))dx=704 + 192 = 896`