Integral of regions

Question

Consider the region $$E = \{(x,y,z)\in \mathbb{R}^3 : x^2+y^2+z^2\leq 1,\; z\leq \sqrt{x^2+y^2}\}$$ How we can draw the region?

How to compute integral limits for $\int_E \frac{z\,x^2}{x^2+y^2}\, dx\,dy\,dz$ ?

Answer 1

HINT

To figure out the region try to consider simple plot on $x-z$, $y-z$ and $x-y$ planes. You should easily find that

• the region $x^2 +y^2 +z^2 \le1$ is the interior of a sphere with radius 1 centered at the origin
• the surface $z=\sqrt{x^2+y^2}$ is a cone on $z>0$ side with vertex in the origin and $z\le\sqrt{x^2+y^2}$ is the region under the cone

To compute the integral a good choice is to use spherical coordinates with

• $x=r\sin\phi\cos\theta$
• $y=r\sin\phi\sin\theta$
• $z=r\cos\phi$

and with the limits

• $0\le \theta \le 2\pi$

• $0\le r \le 1$

• $-\frac{\pi}2\le \phi \le \frac{\pi}4$

Remember also that in this case

$$dx\,dy\,dz=r^2\sin \phi \,d\phi \,d\theta \,dr$$