Calc III
Homework 7
1. A thin plate covers the triangular region bounded by the x-axis and the
lines x = 1 and y = 2x in the …rst quadrant. The plate’s density at the
point (x; y) is (x; y) = 6x + 6y + 6.
(a) Find the center of mass of the plate.
(b) Find the moment of inertia about the x-axis.
2. Assuming that the density function is equal to 1; …nd the center of mass
of the region cut from the …rst quadrant by the circle x2 + y 2 = r2 :
3. Find the center of mass of the in…nite region in the second quadrant enclosed by the coordinate axes and the curve y = ex if the density function
is (x; y) = 1
4. Find the surface area of the surface z = xy that lies within the cylinder
x2 + y 2 = 1:
5. Find the surface area of the part of the sphere centered at the origin of
space with radius 2 that lies inside the paraboloid z = x2 + y 2 : Hint: it
helps to draw a picture …rst.
6. Evaluate the followings
R1R1R1
(a) 0 0 0 (x2 + y 2 + z 2 )dzdydx
R 1 R 1 x2 R 4 x2 y
(b) 0 0
xdzdydx
3
R1 R1 R1 y
7. Given the region of integration of 1 x2 0 dz dy dx; rewrite the integral as an equivalent iterated integral in the order. Do only part a and b
1
8. Find the volume of the solid below
9. Perform the followings integrals in cylindrical coordinates
p
Z2 Z1 Z2
0 0
r2
dz rdr d
0
10. Evaluate the followings integrals in spherical coordinates
Z2 Z=4Z2
0
0
( cos )
0
2
2
sin
d d d