The area of the region enclosed by y4x^2, x^29y, and y4 is equal to:

Question

The area of the region enclosed by $y\le4x^2$, $x^2\le9y$, and $y\le4$ is equal to:

Accepted Answer

For each $y\in[0,4]$, $x$ ranges (on the right) from $\sqrt y/2$ (from $y\le4x^2$) to $3\sqrt y$ (from $x^2\le9y$).

$$\text{Area}=2\int_0^4\left(3\sqrt y-\frac{\sqrt y}2\right)dy=5\int_0^4\sqrt y\,dy=5\cdot\frac23\cdot8=\frac{80}3.$$

Solutions · 1