A Definite Integral

mathcalculus/integrationdifficulty 1/5#integral#polynomial#ftc

Evaluate the definite integral

$$ \int_0^1 x^2 \, dx. $$

Give your answer as an exact fraction. Recall the Fundamental Theorem of Calculus: if
$F'(x) = f(x)$, then $\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$.

Answer & solution

Answer

$ \frac{1}{3} $

Solution

By the power rule $\int x^2\,dx = \dfrac{x^3}{3}$. Evaluating between the limits,
$$\left.\frac{x^3}{3}\right|_{0}^{1} = \frac{1}{3} - 0 = \frac{1}{3}.$$