Spherical and cylindrical problems

Anonymous — Wed, 23 Jul 2014 12:03:06 -0500

I am having a lot of trouble visualizing and understanding how to set up spherical/cylindrical integrals. In particular, I am having trouble with #4 on the homework sheet. "Let $D$ denote the three-dimensional domain above the cone $z=\sqrt{x^2+y^2}$ and inside the sphere $x^2+y^2+z^2<=4$. Evaluate $\int\int\int(x^2+y^2+z^2)dV$." By looking at this, I can guess that spherical coordinates would work well because $x^2+y^2+z^2=P^2$ but past that I am lost. Please help!

In class Problem

Anonymous — Wed, 23 Jul 2014 09:41:37 -0500

In class today my group and I had a little problem solving the second problem written on the board: Evaluate $\int\int\int sin((x^2+y^2+z^2)^{3/2}) dV$ where $D$ is defined as the top half of the solid unit sphere. This is how far we got $$\int_0^\pi\int_0^\pi\int_0^1 (sin((p^2)^{3/2})p^2 sin\phi dP d\phi d\Theta$$ $$\pi\int_0^\pi\int_0^1 sin(p^3)p^2 sin\phi dP d\phi$$ $$u=p^3$$ $$1/3 du=p^2 dP$$ $$\frac{\pi}{3}\int_0^\pi\int_0^1 sin(u) sin\phi du d\phi$$ $$\frac{\pi}{3}\int_0^\pi sin\phi - cos(u) \biggr|_0^1 d\phi$$ $$\frac{\pi}{3}\int_0^\pi 1-cos(1) d\phi$$ $$\frac{\pi}{3} (\phi-\phi cos(1)) \biggr|_0^\pi$$ $$\frac{\pi}{3} (\pi - \pi cos(1) $$ Did we start with the correct domain of integration or did we make a mistake in the calculations?

Mark's Calc III - Individual question feed

Spherical and cylindrical problems

In class Problem