Q:

How to solve for an integral equation from already having the value of the integral?

So I know that $\int_{0}^{m} (mx-x^2)dx$ must equal 8. I also know that $m$ is a positive integer. How do I solve for this without having to use a calculator?

A:

$$8=\int_0^m (mx-x^2) \mathop{dx} = \left[\frac{1}{2}mx^2 - \frac{1}{3}x^3\right]_{x=0}^m = \frac{1}{6} m^3$$

