Winter 2004 Double Integrals in Polar Coordinates – Review MATH 200
1. Evaluate ZZR
y dA where Ris the region in the first quadrant bounded by the circles x2+y2= 4 and
x2+y2= 25.
Comment: Look at R2π
0R5
2(rsin θ)r drdθ.
2. Evaluate ZZDpx2+y2dA where Dis the region bounded by the cardioid r= 1 + cos θ.
Comment: Look at R2π
0R1+cos θ
0r(r dr)dθ.
3. Evaluate ZZD
x dA where Dis the region in the first quadrant that lies between the circles x2+y2= 4 and
x2+y2= 2x..
Comment: Note that x2+y2= 2xis really the circle (x−1)2+y2= 1, with center at (1,0) and radius 1. In
polar coordinates, this circle has the form r= 2 cos θ. Note: at θ= 0, r= 2 while at θ=π/2, r= 0.
Look at: ZZD
x dA =Zπ/2
0Z2 cos θ
r dr dθ.
4. Find the area of the region enclosed by the cardioid r= 1 −sin θ.
Comment: Look at Z ZD
dA =Z2π
0Z1−sin θ
0
r drdθ.
5. Find the area of the region inside the circle r= 3 cos θand outside the cardioid r= 1 + cos θ.
Comment: We need to find the two intersection points: 3 cos θ= 1 + cos θso 2 cos θ= 1 or θ=±π /3. The
desired integral is then Zπ/3
−π/3Z3 cos θ
1+cos θ
r dr dθ.
6. Find the volume of the solid under the cone z=px2+y2and above the ring 4 ≤x2+y2≤25.
Comment:ZZDpx2+y2dA =Z2π
0Z5
2
r(r dr)dθ.
7. Find the volume of the solid that is bounded above by the the sphere x2+y2+z2= 1 and below by z=px2+y2.
Comment: In polar coordinates, the cone is z=rand the sphere is r2+z2= 1. The volume is given by
Z2π
0Z1/√2
0
[p1−r2−r]r drdθ.
8. Find the volume of the solid bounded by the paraboloids z= 3(x2+y2) and z= 4 −(x2+y2).
Comment: In polar coordinates, the paraboloids are z= 3r2and z= 4 −r2. They intersect at 3r2= 4 −r2
or r= 1. We get Z2π
0Z1
0
[(4 −r2)−3r2]rdr dθ.
9. Find the volume of the solid that inside both the cylinder x2+y2= 4 and the ellipsoid 4x2+ 4y2+z2= 64.
Comment: In polar coordinates, the cylinder is r= 2 and the ellipsoid is 4r2+z2= 64. The integral is
Z2π
0Z√64−4r2
−√64−4r2
r dr dθ.
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