Area between curves

A definite integral finds the area under one curve. Area between two curves is the area under the top one, minus the area under the bottom one.

\int_{- 2}^{2} (4 - x^{2}) d x = \frac{32}{3}

is the shaded area.

Work through it

They meet at $x = 0$ and $x = 1$ . On that stretch, $x$ is on top. $\int_{0}^{1} (x - x^{2}) d x$

Integrate. $[\frac{x ^{2}}{2} - \frac{x ^{3}}{3}]_{0}^{1}$

Calculate. $\frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

They meet at $x = \pm 2$ . $\int_{- 2}^{2} (4 - x^{2}) d x$

Integrate and evaluate. $\frac{32}{3}$

Subtracting in the wrong order.

If you get a negative area, you've got bottom minus top.

The short version
Find where the curves meet. Integrate (top minus bottom) between those points.