http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Wed, 23 Jul 2014 11:45:41 -0500http://calc3.askbot.com/question/149/in-class-problem/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?Wed, 23 Jul 2014 09:41:37 -0500http://calc3.askbot.com/question/149/in-class-problem/http://calc3.askbot.com/question/149/in-class-problem/?answer=150#post-id-150The way our group set it up was different, and here is why. The domain is only the top half of the unit sphere, so $\delta\phi$ is only going from zero to $\frac{\pi}{2}$ Since it is all the way around the middle of the sphere, for lack of better wording, $\delta\theta$ will go from zero to $2\pi$. Giving us the following: $$\int_0^\{2\pi}\int_0^\frac{\pi}{2}\int_0^1 (\sin((\rho^2)^{3/2})\\rho^2 \sin\phi d\rho d\phi d\Theta$$ $$\{2\pi}\int_0^\frac{\pi}{2}\int_0^1 \sin(\rho^3)\rho^2 \sin\phi d\rho d\phi$$ Using u sub just like you did... $$u=\rho^3$$ $$1/3 du=\rho^2 d\rho$$ $$\frac{\{2\pi}}{3}\int_0^\frac{\pi}{2}\int_0^1 \sin(u) \sin\phi du d\phi$$ $$\frac{\{2\pi}}{3}\int_0^\frac{\pi}{2} \sin\phi( - \cos(u)) \biggr|_0^1 d\phi$$ $$\frac{\{2\pi}}{3}\int_0^\frac{\pi}{2} \sin\phi(-\cos(1)+1) d\phi$$ $$\frac{\{2\pi}}{3}(-\cos(1)+1) (-\cos(\phi)) \biggr|_0^\frac{\pi}{2}$$ $$\frac{\{2\pi}}{3}(-\cos(1)+1) (1) $$ $$\frac{\{2\pi}}{3}(1-\cos(1)) $$Wed, 23 Jul 2014 11:45:41 -0500http://calc3.askbot.com/question/149/in-class-problem/?answer=150#post-id-150