January 27, 2005 11:45
L24-CH07
Sheet number 1 Page number 290
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CHAPTER 7
Applications of the Definite Integral in Geometry, Science, and Engineering EXERCISE SET 7.1
2
2 (x + 1 − x)dx = (x /3 + x − x /2) = 9/2 2
1. A = −1
4
2. A =
3
2
−1
4 √ ( x + x/4)dx = (2x3/2 /3 + x2 /8) = 22/3
0
0 2
3. A =
2 (y − 1/y 2 )dy = (y 2 /2 + 1/y) = 1
1
4. A =
1 2
2 (2 − y 2 + y)dy = (2y − y 3 /3 + y 2 /2) = 10/3
0
0
4
(4x − x2 )dx = 32/3
5. (a) A =
16
(b) A =
0
√ ( y − y/4)dy = 32/3
0
y (4, 16) y = 4x y = x2 5 x 1
6. Eliminate x to get y 2 = 4(y + 4)/2, y 2 − 2y − 8 = 0, (y − 4)(y + 2) = 0; y = −2, 4 with corresponding values of x = 1, 4.
1
(a) A =
√ √ [2 x − (−2 x)]dx +
0
1
=
√
4 xdx + 0
4
√ [2 x − (2x − 4)]dx
4
√ (2 x − 2x + 4)dx = 8/3 + 19/3 = 9
4
−2
(4, 4)
y2 = 4x
y = 2x – 4 x
1
1
(b) A =
y
[(y/2 + 2) − y 2 /4]dy = 9
290
(1, –2)