Evaluate $\iiint_{R} (2x+y) \,dx \,dy \,dz$

Question

Evaluate $$\iiint _{R} (2x+y) \,dx \,dy \,dz\,,$$ where $R$ is the region bounded by the cylinder $z = 4 - x^{2} $ and the planes $x = 0$, $y = 0$, $y = 2$ and $z = 0$.

How do I extract the limits from the given conditions?

Answer 1

To visualize this shape. Focus on the $x$-$z$ plane first and draw the curve $z=4-x^2$ and then note that $0\le y \le 2$.

$x$ start from value $0$ and when $z=0$, $x=2$. Hence $0 \le x \le 2$.

$z$ is bounded below by $0$ and bounded above by $4-x^2$.

$$\int\int\int_R (2x+y) \, dxdydz = \int_0^2 \int_0^2\int_0^{4-x^2} (2x+y) \, dzdxdy$$

Answer 2

Please note that $z = 4-x^2$ is not a cylinder... In the picture below it is represented the surface $z = 4-x^2$. The planes $x=0$, $y=0$ and $y=2$ are the lateral faces, while $z=0$ is the bottom face.

I think that with this in mind you can place the integration limits.

The final result is $\frac{80}{3}$.