Integration and cylindrical and spherical coordinates

[TheSpaceGuy]

Homework Statement

I have three problems and I could really use some help.
1. Integrate the function f(x,y,z) = y over the part of the elliptic cylinder
x^2/4 +y^2/9 = 1 that is contained in the sphere of radius 4 centered at the origin and such that x≥0, y≥ 0, z≥0.

2. Find the total mass of the half cylinder x^2+y^2 ≤ 9, x≥0, 0≤ z ≤ 2 with density p(x,y,z)= 9-x^2 -y^2 using cylindrical coordinates.

3. Use the spherical coordinates to integrate g(x,y,z) = sqrt(x^2 +y^2 +z^2) over the region x^2 +y^2 +z^3 ≤ 2z. Describe the region geometrically.

The Attempt at a Solution

Basically I just need help setting up the integrals for all of them. So If you can just do that, it would be great.

For the first problem I have so far
int(0 , 4) int(0 , sqrt (9-9x^2 / 4) ) int (0 , 1) y dzdydx
Is this right, and if not what is the correct integral.

For the second problem I have
int( 0 , pi) int(0 , 3) int(0 , 2) 9 - x^2 -y^2 r dzdrdθ
Is this right?

The third problem I have no idea what to do with it.
I know the procedure of spherical so please help me explaining to me how you got the correct integral. Thank you!

 

[HallsofIvy]

The ellipse $x^2/4+ y^2/9$ only extends from x= -2 to x= 2 in the xy-plane so your 'dx' integral should only be from 0 to 2. The limits on the 'dy' integral a correct. The limits on the z-integral should be from 0 up to the sphere, $z= \sqrt{16- x^2- y^2}$. I would have been inclined to put that into cylindrical coordinates. You did that for the second problem, which is correct.

For the third problem, is that really a cube on the z in $x^2+ y^2+ z^3\le 2z$? That would make it very messy!

If it were $x^2+ y^2+ z^2\le 2z$ then you could "complete the square" in z:
$x^2+ y^2+ z^2- 2z+ 1\le 1$ or $x^2+ y^2+ (z- 1)^2\le 1$ which would be the inside and surface of the sphere of radius 1 with center at (0, 0, 1).