Thomas Calculus 13th [Solutions]

CHAPTER 6
APPLICATIONS OF DEFINITE INTEGRALS
6.1 VOLUMES USING CROSS-SECTIONS
1.
2
(diagonal)
x x
A( x) 2 x;
2 2
2
0, 4; b
4 2
4
a b V A ( x
a
) dx
0 2 x dx x 16 0
2.
2 2
2 2 2
2
(diameter)
2 2 1
2 4
A( x) x x x
1 2 x x ; a 1, b 1;
4 4 4
b
1 2 4 2 3
V A( x) dx 1 2x x dx x x x 2 1 2 1 16
a
1 3 5
1
3 5 15
5 1
3.
2 2
2 2 2 2 2
A( x) (edge) 1 x 1 x 2 1 x 4 1 x ; a 1, b 1;
b
1 2
V A( x) dx 4 1 x dx 4 x x 8 1 1 16
a
1 3
1
3 3
3 1
4.
2 2
2 2 2
2 1 1 2 1
(diagonal) x x 2
A( x) 2 x
2 1 x ; a 1, b 1;
2 2 2
b
1 2
V A( x) dx 2 1 x dx 2 x x 4 1 1 8
a
1 3
1
3 3
3 1
5. (a) STEP 1) A( x) 1 (side) (side) sin 1 2 sin x 2 sin x sin 3 sin x
2 3 2 3
STEP 2) a 0, b
b
STEP 3) V A( x) dx 3 sin x dx 3 cos x 3(1 1) 2 3
a
0 0
2
(b) STEP 1) A( x) (side) 2 sin x 2 sin x 4 sin x
STEP 2) a 0, b
STEP 3)
b
V A ( x ) dx
a
0
4 sin x dx 4cos x
0
8
2
(diameter) 2 2 2
6. (a) STEP 1) A( x) (sec x tan x) sec x tan x 2sec x tan x
4 4 4
2 2
sec x sec x 1 2 sin x
4 2
cos x
STEP 2) a , b
3 3
STEP 3) V b
( ) /3 2
/3
2sin 1
/3 4 2sec 1 x
a
A x dx x dx 2
cos 4 2 tan x x 2
x
cos x /3
2 3 2 1 2 3 2 1 4 3 2
4 3 3 4 3
1 1
2 2
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