Question

Calc 3 Spherical Coordinates Help
Integrate the function f(x,y,z)=−6x+8y over the solid given by the figure below, if P=(-7,-7,2), Q=(-4,-4,2), R=(4,4,2), and S=(7,7,2).

Answer 1

\[\int\limits_{0}^{4}\int\limits_{0}^{\pi}\int\limits_{0}^{\pi/2}-6(\rho \cos \theta \sin \theta) + 8 (\rho \sin^2 \theta )(\rho^2 \sin \phi d \rho d \phi d \theta) \]

Answer 2

Would that be correct?

Answer 3

you should be your d rho d phi d theta in the order to match up with the integral signs
i.e. what is the inner most integral ?

Answer 4

btw, cylindrical coordinates seem nicer for this problem. do you have to use spherical ?

Answer 5

the radius should go from 4 sqr(2) to 7sqr(2)

Answer 6

theta goes from -3pi/2 to + pi/2
z from 0 to 2

Answer 7

I don't know why I've been trying spherical, I should have tried cylindrical.

Answer 8

Why'd you say the radius goes to 7 sqrt(2)?

Answer 9

Ah, I understand

Answer 10

point S (for example) has (x,y) coords (7,7)
the distance from 0,0 to 7,7 is 7 sqr(2)

Answer 11

in case it's not obvious, P,Q,R,S all lie on the same line

Answer 12

I understand.
I'm trying to evaluate the integral through WolframAlpha (Yes, I understand I'm awful).
But it seems as if can't get the function to work.
Are my bounds correct?
http://bit.ly/1GF76O7

Answer 13

yes. it seems the answer is 0 ?

Answer 14

That can't be correct.
I'll manually do it.

Answer 15

oh, I think theta should go from -3pi/4 to pi/4 (not over 2)

Answer 16

Over 4?

Answer 17

You're correct, and after looking at the unit circle and comparing it to the picture, I understand.
Thanks so much!!!