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CHAPTER 44
44.19
Find the volume V of the tetrahedron bounded by the coordinate planes and the plane z = 6 — 2x + 3y.
As shown in Fig. 44-8, the solid lies above the triangle in the ry-plane bounded by 2x + 3y = 6 and
the x and y axes.
(Check against the formula
Fig. 44-8
Fig. 44-9
44.20
44.21
Use a double integral to find the area of the region Si bounded by xy = 1 and 2x + y = 3.
Figure 44-9 shows the region St.
Find the volume V of the solid bounded by the right circular cylinder x
2 + y
2 = 1, the ry-plane, and the plane
x + z = 1.
As seen in Fig. 44-10, the base is the circle x
2 + y
2 = I in the ry-plane, the top is the plane x + z = 1.
(Note: We know that
since the integral is the area of the unit semicircle.)
Fig. 44-10
44.22
44.23
Find the volume V of the solid bounded above by the plane z = 3x + y + 6, below by the ry-plane, and on the
sides by y = 0 and y = 4 - x
2 .
Since -2<.x<2 and y^O, we have z = 3x + y + 6^0. Then
Find the volume of the wedge cut from the elliptical cylinder 9x
2 + 4y
2 = 36 by the planes z = 0 and
z = y+ 3.
On 9x
2 + 4y
2 = 36, -3<>>s3. Hence, z = y + 3>0. So the plane z = >> + 3 will be above the
plane z =0 (see Fig. 44-11). Since the solid is symmetric with respect to the yz-plane, V =
dy represents the area of the upper semicircle
[The integral
of the circle x
2 + y
2 = 9. Hence, it is equal to