Integration over solid regions

[Daniel_Feldman]

1. I need to integrate the function f(x,y,z)=-8x+2y over this solid. P=(1,5,0) and Q=(-1,5,4).

I need help in setting up the limits of integration.

2. I need to integrate the same function over the solid above z=0 contained between spheres centered on the origin with radii 4 and 5, excluding everything in the first octant.

I think I need to set this up in spherical coordinates, but again, i need help setting up my limits.

 

[galactus]

2. I need to integrate the same function over the solid above z=0 contained between spheres centered on the origin with radii 4 and 5, excluding everything in the first octant.

Let's see, I will give it a go. See what you think.

\(\displaystyle \L\\x={\rho}sin({\phi})cos({\theta}), \;\ y={\rho}sin({\phi})sin({\theta})\)

This gives: \(\displaystyle \L\\{\rho}^{2}sin({\phi})(-8({\rho}sin({\phi})cos({\theta}))+2(={\rho}sin({\phi})sin({\theta})))=-2{\rho}^{3}(4cos({\theta})-sin({\theta}))sin^{2}({\phi})\)

\(\displaystyle \L\\\int_{0}^{2{\pi}}\int_{0}^{\pi}\int_{4}^{5}-2{\rho}^{3}(4cos({\theta})-sin({\theta}))sin^{2}({\phi})d{\rho}d{\phi}d{\theta}\)

Now, you needed all but the first quandrant. Subtract 1/8th of what you get from it.

 

[soroban]

Hello, Daniel!

1. I need to integrate the function \(\displaystyle f(x,y,z)\:=\:-8x\,+\,2y\) over this solid.
\(\displaystyle P\,=\,(1,5,0)\) and \(\displaystyle Q\,=\,(-1,5,4)\)

I recommend cylindrical coordinates.

The solid is a "slice of cake".
. . Its radius is: \(\displaystyle \,r\,=\,\sqrt{26}\)
. . Its central angle ranges from: \(\displaystyle \,\theta\,=\,\arctan(-5)\,\) to \(\displaystyle \,\theta\,=\,\arctan(5)\)
. . And \(\displaystyle z\,=\,0\) to \(\displaystyle z\,=\,4\)