a). For this part I got an ice cream cone shape. I tried drawing it on Matlab, but I kept getting red error messages!Let $\displaystyle \Omega$ be the three dimensional region defined by $\displaystyle \Omega:=\{(x,y,z)| z\geq 0, x^2+y^2 \leq z^2 \leq 1-x^2-y^2 \}$.

a). Sketch the region $\displaystyle \Omega$

b). Evaluate $\displaystyle \int \int \int_{\Omega} dx \ dy \ dz$

Hint: It may be helpful to use spherical polar coordinates.

This ice cream cone has $\displaystyle 0 \leq z \leq 1$ and $\displaystyle x^2+y^2= \left( \frac{1}{\sqrt{2}} \right)^2$.

I found this helpful for part b).!

b). Is the required integral $\displaystyle 2 \int_0^{\frac{1}{\sqrt{2}}} \int_0^z \int_{0}^y \ z \ dx \ dy \ dz+ \frac{1}{2} \int_0^{\frac{\pi}{2}} \int_0^{2 \pi} \int_0^{\frac{1}{\sqrt{2}}} \ r^3 sin (\phi) \ dr \ d \theta \ d \phi$?

Also, is there an easier way to write this integral???