M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY
Interphase Calculus III Problem Set 4
Instructor: Samuel S. Watson
Due: August 3, 2015
1. Integrate ( x + y)2 over the triangular region with vertices at the origin, (3, 0), and (0, 4).
2. Find the volume of a hemisphere of radius R by integrating
centered at the origin.
q
R2 − x2 − y2 over the disk of radius R
3. Consider the region R between the parabolas y = 1 − x2 and y = x2 − 7. Find
4. Find
Z 1 Z √1− x 2
√
−1 − 1− x 2
ZZ
R
xy dA.
1
dy dx by switching to polar coordinates.
( x2 + y2 )1/10
5. Find the volume of the region W that represents the intersection of the solid cylinder x2 + y2 ≤ 1 and the
solid ellipsoid 2( x2 + y2 ) + z2 ≤ 10.
6. Evaluate
Z 3 Z √9−y2 Z √18− x2 −y2
0
√
0
x 2 + y2
x2 + y2 + z2 dz dx dy
by rewriting the integral in spherical coordinates.
7. Find the surface area of the part of the sphere x2 + y2 + z2 = 4 that lies above the plane z = 1.
y
dA where R is the region bounded by the curves y = 0, y = x/2, x2 − y2 = 1, and x2 − y2 = 4,
x
by changing coordinates. (Hint: try letting y/x be one of your new coordinates.)
8. Find
ZZ
R
9. The Jacobian of a transformation
x = g(u, v, w),
y = h(u, v, w),
z = k(u, v, w).
is given by

∂x

 ∂u

 ∂y
det 

 ∂u

 ∂z
∂u
∂x
∂v
∂y
∂v
∂z
∂v

∂x

∂w 

∂y 


∂w 

∂z 
.
∂w
Confirm that for the cylindrical coordinate transformation, that is (u, v, w) = (r, θ , z), the Jacobian is equal
to r. Confirm that for the spherical coordinate transformation, that is (u, v, w) = (ρ , θ , φ), the Jacobian is
equal to ρ 2 sin φ.