558
Chapter 5 | Integration
First, eliminate the radical by rewriting the integral using rational exponents. Then, separate the numerator terms by writing each one over the denominator: ⌠ ⌡ 1 9 x −1 x 1/2 dx = ⌠ ⌡ 1 9 ⎛ ⎝ x x 1/2 − 1 x 1/2 ⎞ ⎠ dx . Use the properties of exponents to simplify: ⌠ ⌡ 1 9 ⎛ ⎝ x x 1/2 − 1 x 1/2 ⎞ ⎠ dx = ∫ 1 9 ⎛ ⎝ x 1/2 − x −1/2 ⎞ ⎠ dx . Now, integrate using the power rule:
⎞ ⎠ ⎟ | 1 9
⎛ ⎝ ⎜ x 3/2 3 2
9 ⎛ ⎝ x 1/2 − x −1/2 ⎞
1/2 1 2
− x
∫
⎠ dx =
1
⎡ ⎣ ⎢ (9) 3/2 3 2
⎤ ⎦ ⎥ −
⎡ ⎣ ⎢ (1) 3/2 3 2
⎤ ⎦ ⎥
1/2
1/2
=
− (9)
− (1)
1 2
1 2
⎡ ⎣ 2 3
(27)−2(3) ⎤
⎡ ⎣ 2 3
(1)−2(1) ⎤ ⎦
⎦ −
=
=18−6− 2 3 +2 = 40 3 .
See Figure 5.29 .
Figure 5.29 The area under the curve from x =1 to x =9 can be calculated by evaluating a definite integral.
Use The Fundamental Theorem of Calculus, Part 2 to evaluate ∫ 1 2 x −4 dx .
5.19
Example 5.22 A Roller-Skating Race
James and Kathy are racing on roller skates. They race along a long, straight track, and whoever has gone the
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