Practice Test Chapter 121. Use double
integration to find the center of mass of the region x^2 + y^2 <= 9^2, y >=
0, if the density function is delta(x,y) = 5 * (x^2 + y^2)
2. Evaluate the triple integral of x^2*y over D with respect to V where D
is the tetrahedron with vertices (0,0,0), (1,0,0), (0,4,0), (0,0,2).
3. Find the center of mass of the tetrahedron in the first octant bounded by
the plane x/a + y/b + z/c where a,b,c > 0. Assume density function delta =
x.
4. Evaluate the double integral Int( Int((2x + 3y) dy), -1, x) dx, -1, 1).
5. Use spherical coordinates to compute the triple integral of z / (x^2 +
y^2 + z^2) with respect to V over D where D is the solid bounded above by
the sphere x^2 + y^2 + z^2 = 4 and below by the surface phi = pi/6.
6. Show that a homogenous lamina of mass m that covers the circular region
x^2 + y^2 = a^2 will have moment of inertia ma^2 / 4 with respect to both
the x- and y-axes.
7. Use a double integral to find the area of the region described by 0 <= r
<= 3 sin (2 theta).
8. Convert the equation 3x^2 + 3y^2 + 3z^2 = 1 to cylindrical coordinates.
9. Find the surface area of the portion of the surface z = x^2 that lies
above the triangular region in the plane with vertices (0,0,0), (1,0,0) and
(0,1,0).
10. Set up a double integral for the volume of the solid remaining when an
off-center square hole with side 1, with one edge of the hole passing
through the center of the sphere, is drilled through a sphere with radius 2.