Math 251 Week in Review 13.4, 13.5, 13.6
1. Sketch each polar equation in the x,y-plane.
a)
r  cos( 2 )
c)
r  2  cos 
b)
b)
r  cos( 3 )
r
1
 cos 
2
2. Find a Cartesian equation for the given polar equation.
1
1  sin 
a)
r
d)
r  3 csc
b)
r2  
c)
r  2 sin 
3. Find a polar equation for each Cartesian equation.
a)
x2  y2  4
b)
xy  1
c)
x2  4 y
4. Evaluate each double integral by changing to polar coordinates.
a)
 y dA
R is bounded by x 2  y 2  16
yx
y0
R
in the 1st quadrant.
b)
 sin(
x 2  y 2 ) dA
R is the region 4  x 2  y 2  16
y  0.
R
c)
 xdA
R is the region between x 2  y 2  2 x
and
x 2  y 2  16
R
5. Find the area of the region in the x,y-plane enclosed by each polar graph.
a)
r  1  sin 
b)
inside r  4 sin 
and outside r  2
6. Use polar coordinates to find the volume of each region.
z  3 x 2  3 y 2 and z  4  x 2  y 2
a) the region bounded by
b) the region under the plane 6x+2y+3z=18 and over the disk x  y  2 x
2

7. Use polar coordinates to show that
2
x
 e dx   .
2

8. Evaluate each double integral by converting to polar coordinates.
2
3 9 y
a) a )
 
0
(x  y )
2
2 32
2 2x x2
dxdy
0
b)

 (x
0
0
2
 y 2 )1 2 dydx
9. Find the mass and center of mass of:
a) the triangle with corners (0,0), (1,1), and (4,0) if the density function is ρ(x,y)=x.
b) the region inside x  y  2 y and outside x  y  1 if the density function is
2
 ( x, y ) 
k
x y
2
2
2
.
2
2