3.3. MULTIPLE INTEGRALS 113
Problem 3.50 Find the mass of the plate 0 x; y 3 if
.x; y/ D y
2
C 1:
Problem 3.51 Find the center of mass of the region bounded by the x-axis, the y-axis, and
the line x C y D 4 if the density function is
.x; y/ D y C 1:
Problem 3.52 Find the center of mass of the region bounded by y D
p
x and y D x
2
if the
density function is
.x; y/ D x C2:
Problem 3.53 Find the center of mass of the region 0 x; y 1 if the density function is
.x; y/ D .x Cy/=2:
Problem 3.54 Find the volume under
f .x; y/ D
p
x
2
C y
2
above the region x
2
C y
2
16.
Problem 3.55 Find the volume under
f .x; y/ D x
2
C y
2
above the region bounded by the petal curve r D 2 cos.3/.
Problem 3.56 Find the volume under
f .x; y/ D .x
2
C y
2
/
3=2
above the region bounded by the petal curve r D cos.2 /.
114 3. ADVANCED INTEGRATION
Problem 3.57 Find a plane f .x; y/ so that the area under the plane but over a circle of radius
2 centered at the origin is 16 units
3
.
Problem 3.58 Derive the general formula for the volume over a rectangle and under a plane
in a region where the plane has a positive
z
value.
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