Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

268 Multiple and Line Integrals [Ch. 7
2258. Passing to spherical coordinates, evaluate the integral
where V is the interior of the sphere x 2
(V)
2
-\-y +z 2
B. Computing volumes by means of triple integrals
2259. Use a triple integral to compute the volume of a solid
the surfaces
bounded by
2260**. Compute the volume of that part of the cylinder
x 2
-f tf = 2
2ax which is contained between the paraboloid* + y 2 = 2az
and the xy-plane.
2261*. Compute the volume of a solid bounded by the sphere
x 2
+y 2
+z 2 =a 2
and the cone z 2
---x 2
+ /y" (external to the cone).
2262*. Compute the volume of a solid bounded by the x
sphere
2
+y z
+z 2 = 4 and the paraboloid x 2
+if=-3z (internal to the
paraboloid).
2263. Compute the volume of a solid bounded by the xy-plane,
the cylinder x 2
+y z = ax and the sphere x 2
+y 2
-f- z z = a 2
(internal
to the cylinder).
2264. Compute the volume of a solid bounded by the paraboloid
Fig. 100
and the plane -+- = 1 (a
- + -~ = 2 -i and the plane x--=a.
C. Applications of triple integrals
to mechanics and physics
2265. Find the mass M of a rectangular
parallelepiped Q^x^a,