Chapter 5 | Integration
559
farthest after 5 sec wins a prize. If James can skate at a velocity of f ( t ) =5+2 t ft/sec and Kathy can skate at a velocity of g ( t ) =10+cos ⎛ ⎝ π 2 t ⎞ ⎠ ft/sec, who is going to win the race?
Solution We need to integrate both functions over the interval ⎡ ⎣ 0, 5 ⎤
⎦ and see which value is bigger. For James, we want to
calculate
5 (5+2 t ) dt .
∫
0
Using the power rule, we have
∫ ⎞ ⎠ | 0 5 = (25 + 25) = 50. Thus, James has skated 50 ft after 5 sec. Turning now to Kathy, we want to calculate ∫ 0 5 10+cos ⎛ ⎝ π 2 t ⎞ ⎠ dt . Weknow sin t is an antiderivative of cos t , so it is reasonable to expect that an antiderivative of cos ⎛ ⎝ π 2 t ⎞ ⎠ would involve sin ⎛ ⎝ π 2 t ⎞ ⎠ . However, when we differentiate sin ⎛ ⎝ π 2 t ⎞ ⎠ , we get π 2 cos ⎛ ⎝ π 2 t ⎞ ⎠ as a result of the chain rule, so we have to account for this additional coefficient when we integrate. We obtain ∫ 0 5 10+cos ⎛ ⎝ π 2 t ⎞ ⎠ dt = ⎛ ⎝ 10 t + 2 π sin ⎛ ⎝ π 2 t ⎞ ⎠ ⎞ ⎠ | 0 5 = ⎛ ⎝ 50+ 2 π ⎞ ⎠ − ⎛ ⎝ 0− 2 π sin0 0 5 (5+2 t ) dt = ⎛ ⎝ 5 t + t 2 ⎞ ⎠
≈50.6. Kathy has skated approximately 50.6 ft after 5 sec. Kathy wins, but not by much!
5.20 Suppose James and Kathy have a rematch, but this time the official stops the contest after only 3 sec. Does this change the outcome?
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