Thomas Calculus 13th [Solutions]

Section 6.1 Volumes Using Cross-Sections 433
11. The slices perpendicular to the edge labeled 5 are triangles, and by similar triangles we have b 4 h 3 b
h 3 4 .
The equation of the line through (5, 0) and (0, 4) is y 4 x 4, thus the length of the base 4 x 4 and
5
5
the height 3 4 x 4 3 x 3. Thus A( x) 1 (base) (height) 1 4 x 4 3 x 3
4 5 5
2 2 5 5
6 2 12
b
5 2 3 2
5
x x 6 and V A
6 12 2 6
25 5
( x ) dx x x
0 25 5 6 dx x x
25 5
6 x (10 30 30) 0 10
a
0
12. The slices parallel to the base are squares. The cross section of the pyramid is a triangle, and by similar
triangles we have
b 3
b
3
h Thus
3
5
3
y
25 0
h
15 0 15
5 5 .
2 3
2
9 2 d
5
9 2
5 25 c
0 25
A( y) (base) y y V A( y)
dy y dy
13. (a) It follows from Cavalieris Principle that the volume of a column is the same as the volume of a right
prism with a square base of side length s and altitude h. Thus,
2 2
STEP 1) A( x) (sidelength) s ;
STEP 2) a 0, b h;
STEP 3)
b
h 2 2
V A( x)
dx s dx s h
a
0
(b) From Cavalieris Principle we conclude that the volume of the column is the same as the volume of the
2
prism described above, regardless of the number of turns V s h
14. 1) The solid and the cone have the same altitude
of 12.
2) The cross sections of the solid are disks of
diameter x x x
If we place the vertex of
2 2 .
the cone at the origin of the coordinate system
and make its axis or symmetry coincide with
the x -axis then the cones cross sections will
be circular disks of diameter x x x
4 4 2
(see accompanying figure).
3) The solid and the cone have equal altitudes and
identical parallel cross sections. From
Cavaliers Principle we conclude that the solid
and the cone have the same volume.
15.
16.
17.
2 2 2 2 2
R ( x ) y 1 x V R
2 0 ( x ) dx
0 1 x dx
2 0 1 x x dx x x x
4 2 12
0
2 4 8 2
2 12 3
2 2 3 2
( ) 3y
2 2 2 3
2 2 2 3
2
9 3 3
2 0 ( ) y
R y x V R y dy dy y dy y
0 2 0 4 4 0 4
8 6
R( y) tan y
4
; u y
4
du dy
4
4 du dy;
2
y 0 u 0, y 1 u ;
4
1 2 1 /4 2 /4 2
/4
V R
0 ( y ) dy
0 tan y dy
4
4 0 tan u du 4 0
1 sec u du 4 u tan u
0
4 1 0 4
4
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