- #1
jmm5872
- 43
- 0
Make the indicated change of variables (do not evaluate) (Not sure how to write an iterated integral with bounds so I will try and explain by just writing the bounds)
(I also tried using the symbols provided, but everything I tried just put a theta in here so I gave up)
[tex]\int\int\int[/tex]xyz dzdxdy
-1 < z < 1
-sqrt(1-y^2) < x < sqrt(1-y^2) ... circle with radius 1
-sqrt(4-x^2-y^2) < y < sqrt(4-x^2-y^2) ... sphere with radius 4
Relevant equations:
x = rcos[tex]\theta[/tex]
y = rsin[tex]\theta[/tex]
z = z
Attempt:
I understand the bounds, it is a sphere with a radius of 4, however, the bounds constrain this to a cylinder with a radius of 1 along the z-axis. This makes it a cylinder with a rounded top and bottom (the only part of the sphere left).
My initial change of the bounds was this:
-4 < z < 4
0 < r < 1
0 < theta < 2pi
However I know this is wrong because these bounds only make a cylinder with radius 1 and length 8, they do not account for the rounded "caps."
My only idea was that I would have to split into two separate integrals, one for the cylinder and one for the top cap (double that one due to symmetry). But it seems like there should be a better way to account for the endcaps.
Thanks
(I also tried using the symbols provided, but everything I tried just put a theta in here so I gave up)
[tex]\int\int\int[/tex]xyz dzdxdy
-1 < z < 1
-sqrt(1-y^2) < x < sqrt(1-y^2) ... circle with radius 1
-sqrt(4-x^2-y^2) < y < sqrt(4-x^2-y^2) ... sphere with radius 4
Relevant equations:
x = rcos[tex]\theta[/tex]
y = rsin[tex]\theta[/tex]
z = z
Attempt:
I understand the bounds, it is a sphere with a radius of 4, however, the bounds constrain this to a cylinder with a radius of 1 along the z-axis. This makes it a cylinder with a rounded top and bottom (the only part of the sphere left).
My initial change of the bounds was this:
-4 < z < 4
0 < r < 1
0 < theta < 2pi
However I know this is wrong because these bounds only make a cylinder with radius 1 and length 8, they do not account for the rounded "caps."
My only idea was that I would have to split into two separate integrals, one for the cylinder and one for the top cap (double that one due to symmetry). But it seems like there should be a better way to account for the endcaps.
Thanks