VOLUME 0 177
22.21
The solid is a wedge, cut from a perfectly round tree of radius r by two planes, one perpendicular to the axis of the
tree and the other intersecting the first plane at an angle of 30° along a diameter. (See Fig. 22-13.)
Fig. 22-14
Fig. 22-15
22.24
Let 91 be the region between y = x
3
,
region 9? about v = — 1.
x = \, and y = 0. Find the volume of the solid obtained by rotating
The same volume is obtained by rotating about the *-axis (y = 0) the region obtained by raising 3$ one unit,
that is, the region bounded by y = x
3 + l, y = l, and x. = 1 (see Fig. 22-16). By the circular ring
formula, this is
22.23
The tetrahedron formed by three mutually perpendicular edges of lengths a,b,c.
Let the origin be the intersection of the edges, and let the jc-axis lie along the edge of length c (Fig. 22-15). A
typical cross section is a right triangle with legs of lengths d and e, parallel respectively to the edges of lengths a
and b. By similar triangles,
By the cross-section formula
and
So, the area
22.22
A square pyramid with a height of h units and a base of side r units.
Locate the x-axis perpendicular to the base, with the origin at the center of the base (Fig. 22-14). By similar
right triangles,
formula,
and
So,
and, by the cross-section
Let the x-axis be the intersection of the two planes, with the origin on the tree's axis. Then a typical cross
section is a right triangle with base
and height
So, the area A
is
By symmetry, we can compute the volume for x > 0 and
then double the result. The cross-section formula yields the volume
22.21
The solid is a wedge, cut from a perfectly round tree of radius r by two planes, one perpendicular to the axis of the
tree and the other intersecting the first plane at an angle of 30° along a diameter. (See Fig. 22-13.)
Fig. 22-14
Fig. 22-15
22.24
Let 91 be the region between y = x
3
,
region 9? about v = — 1.
x = \, and y = 0. Find the volume of the solid obtained by rotating
The same volume is obtained by rotating about the *-axis (y = 0) the region obtained by raising 3$ one unit,
that is, the region bounded by y = x
3 + l, y = l, and x. = 1 (see Fig. 22-16). By the circular ring
formula, this is
22.23
The tetrahedron formed by three mutually perpendicular edges of lengths a,b,c.
Let the origin be the intersection of the edges, and let the jc-axis lie along the edge of length c (Fig. 22-15). A
typical cross section is a right triangle with legs of lengths d and e, parallel respectively to the edges of lengths a
and b. By similar triangles,
By the cross-section formula
and
So, the area
22.22
A square pyramid with a height of h units and a base of side r units.
Locate the x-axis perpendicular to the base, with the origin at the center of the base (Fig. 22-14). By similar
right triangles,
formula,
and
So,
and, by the cross-section
Let the x-axis be the intersection of the two planes, with the origin on the tree's axis. Then a typical cross
section is a right triangle with base
and height
So, the area A
is
By symmetry, we can compute the volume for x > 0 and
then double the result. The cross-section formula yields the volume