Q.Evaluate $\displaystyle\int_0^2 x^2\,dx$.

The power-rule antiderivative evaluated at the two limits gives the definite integral directly.

Finding the antiderivative

Applying the Fundamental Theorem

$\int_0^2x^2\,dx=F(2)-F(0)=\frac{2^3}{3}-\frac{0^3}{3}=\frac83-0=\frac83$ …

Evaluating $F(0)$ carelessly as something other than exactly $0$ for a pure power function — always substitute the …