# Homework Help: Double integral and polar cordinates other problem.

1. Nov 16, 2012

### christian0710

If we have to find the volume, written in polar cordinates, inside this sphere X2+y2+z2=16 and outside this cylinder x2+y2=4

How should I approach this?
Could I take the sphere function and reqrite in polar cordinates z=√(16-X2-y2) which is the same as z=√(16-r2)

But then I have to make r depend on a function of the cylinder right?

so x2+y2=4 ---> r2=4 so r=2 this must be the boundaries of the cylinder..

Now I get a bit confused. Do we subtract the sphere function from the cylinder function? Or do we make the cylinder function a function that r depends on?

2. Nov 16, 2012

### SammyS

Staff Emeritus
What does r have to be to be outside the cylinder?

3. Nov 17, 2012

### christian0710

Hmm then this would make sense 2<r<4 because the cylinder has a radius of 2, so i guess the theta would go from 0 to 2pi ?

4. Nov 17, 2012

### SammyS

Staff Emeritus
Yes.

What is the function you integrate, i.e., what is your integrand?

5. Nov 17, 2012

### christian0710

The function we want to integrate must be the sphere, right?
X^2+y^2+z^2=16 so if we isolate z we must get z=√(16-X^2-y^2) and written in polar form
z=√(16-r^2) So this is the function we integrate?

6. Nov 17, 2012

### SammyS

Staff Emeritus
That's the equation for the top half of the sphere, the half for which z is positive.

7. Nov 17, 2012

### christian0710

I see, so if 2<r<4 and 0< theta< 2pi
Hmm so we could add two double integrals? one of the lower, and one of the upper sphere?

8. Nov 17, 2012

### SammyS

Staff Emeritus
... or multiply the first integral by 2, because of the symmetry involved.

Do you know what dxdy , the element of area, is in polar coordinates?

9. Nov 17, 2012

### christian0710

yes that's a smart solution :) The polar form of dxdy must be r*dr*dθ Right?

I assume it means the rectangles delta r times delta θ multiplied by the length r (a bit hard for me to visualize)

10. Nov 17, 2012

### christian0710

So 2*∫(∫((√(16-r^2) )*r,r,2,4),θ,0,2∏) This must be the solution? :)

11. Nov 17, 2012

### SammyS

Staff Emeritus
2*∫∫(√(16-r^2) )*r*dr*dθ; where r goes from 2 to 4, θ goes from 0 to 2π .

In my opinion, you should always include the differentials when writing an integral.

12. Nov 18, 2012

### christian0710

Ahh yes a good idea. Thank you very much for your help :)

13. Nov 18, 2012

### christian0710

Hmm but is there an easier way? Because (√16-r2)r is not that easy to do by hand, and id love to be able to do this by hand :)

14. Nov 18, 2012

### SammyS

Staff Emeritus
[STRIKE]Multiply through by r .[/STRIKE]

Putting the parentheses in the correct place gives
( √(16-r2) )r​

Do integration by substitution.

What is the derivative of 16-r2 ?

Last edited: Nov 18, 2012