Calculus Answers

Use spherical coordinates to evaluate the triple integral
∭E (e^(−(x^2+y^2+z^2)))/√(x^2+y^2+z^2) dV,
where E is the region bounded by the spheres x^2+y^2+z^2=4 and x^2+y^2+z^2=9.

Find the volume of the ellipsoid x^2+y^2+5z^2=64.

Use spherical coordinates to evaluate the triple integral ∫∫∫E x^2+y^2+z^2 dV, where E is the ball: x^2+y^2+z^2≤16.

Evaluate the triple integral of f(x,y,z)=z(x^2+y^2+z^2)^(−3/2) over the part of the ball x^2+y^2+z^2≤16 defined by z≥2.

Use cylindrical coordinates to calculate ∫∫∫W f(x,y,z) dV for the given function and region:

f(x,y,z)=z, x^2+y^2≤z≤16

Evaluate the triple integral ∭E z dV where E is the solid bounded by the cylinder y^2+z^2=576 and the planes x=0,y=4x and z=0 in the first octant.

Evaluate the triple integral ∭E x dV where E is the solid bounded by the paraboloid x=5y^2+5z^2 and x=5.

Match the integrals with the type of coordinates which make them the easiest to do. Put the letter of the coordinate system to the left of the number of the integral.

1. ∫(0 to 1)∫(0 to y^2) 1/x dx dy
2. ∫∫∫E dV where E is: x^2+y^2+z^2≤4, x≥0, y≥0, z≥0
3. ∫∫∫E z^2 dV where E is: −2≤z≤2, 1≤x^2+y^2≤2
4. ∫∫∫E z dV where E is: 1≤x≤2, 3≤y≤4, 5≤z≤6
5. ∫∫D 1/(x^2+y^2) dA where D is: x^2+y^2≤4

A. cylindrical coordinates
B. cartesian coordinates
C. polar coordinates
D. spherical coordinates

Evaluate the following integral by converting to spherical coordinates: ∫(0 to 1)∫(0 to sqrt(1-x^2))∫(sqrt(x^2+y^2) to sqrt(2-x^2-y^2)) xy dz dy dx

Evaluate ∫∫∫D e^(sqrt(x^2+y^2+z^2)) dV, where D is the region enclosed by x^2+y^2+z^2= 4 within the first octant. Recall that the first octant is the region for which x >0; y >0; z >0.