The area (in sq.

Question

The area (in sq. units) of the region described by $$
\left\{(x, y): y^2 \leq 2 x \text {, and } y \geq 4 x-1\right\}
$$ is

Accepted Answer

$$\text { Area }=\int_\limits{-\frac{1}{2}}^1\left(\frac{y+1}{4}-\frac{y^2}{2}\right) d y$$

$$\begin{aligned}
& =\left[\frac{y^2}{8}+\frac{y}{4}-\frac{y^3}{6}\right]_{-\frac{1}{2}}^1 \
& =\left(\frac{1}{8}+\frac{1}{4}-\frac{1}{6}\right)-\left(\frac{1}{32}-\frac{1}{8}+\frac{1}{48}\right) \
& =\frac{5}{24}+\frac{7}{96} \
& =\frac{27}{96} \
& =\frac{9}{32}
\end{aligned}$$

Solutions · 1