44.72
44.71
Evaluate
Fig. 44-39
By Problem 44.71,
where 91 is the rectangular box: ls^<2, 0<^<1, 0 If 91 is a rectangular box x l 44.70
44.69
Fig. 44-37
Fig. 44-38
Describe the solid whose volume is given by the integral
dz dy dx.
See Fig. 44-38. The solid lies under the plane z = 3 - x — y and above the triangle in the ry-plane
bounded by the coordinate axes and the line * + y = 3. It is a tetrahedron, of volume
Describe the solid whose volume is given by the integral
dz dy dx, and compute the volume.
The solid is the part of the solid right circular cylinder x
2 + y
2 < 25 lying in the first octant between z = 0
and z = 3 (see Fig. 44-39). Its volume is
MULTIPLE INTEGRALS AND THEIR APPLICATIONS
419
The projection of the intersection of the surfaces is the ellipse 4x
2 + 9y
2 = 36. By symmetry, we can
calculate the integral with respect to x and y in the first quadrant and then multiply it by 4.
Let
x = 3 sin 6,
dx = 3 cos 6 d0.
Then
(from Problem 44.29).
44.68
Describe the solid whose volume is given by the integral
See Fig. 44-37. The solid lies under the plane 2 = 1 and above the region in the first quadrant of the
xy-plane bounded by the lines y = 2x and y = 4.