700
Chapter 6 | Applications of Integration
b ρw ( x ) s ( x ) dx
F = ∫
a
540
x ⎞ ⎠ ( x −10) dx =62.4 ∫ 10 540
⎛ ⎝ 1250− 2 3
⎡ ⎣ x 2 −1885 x +18750 ⎤
= ∫
62.4
− 2 3
⎦ dx
10
⎦ | 10
⎡ ⎣ x 3
x ⎤
540
⎛ ⎝ 2 3
⎞ ⎠
1885 x 2
=−62.4
≈ 8,832,245,000 lb = 4,416,122.5 t.
3 −
2 +18750
Note the change from pounds to tons (2000 lb = 1 ton) (step 4). b. Notice that the drought changes our depth function, s ( x ), and our limits of integration. We have s ( x ) = x −135. The lower limit of integration is 135. The upper limit remains 540. Evaluating the integral, we get F = ∫ a b ρw ( x ) s ( x ) dx = ∫ 135 540 62.4 ⎛ ⎝ 1250− 2 3 x ⎞ ⎠ ( x −135) dx =−62.4 ⎛ ⎝ 2 3 ⎞ ⎠ ∫ 135 540 ( x −1875)( x −135) dx =−62.4 ⎛ ⎝ 2 3 ⎞ ⎠ ∫ 135 540 ⎛ ⎝ x 2 −2010 x +253125 ⎞ ⎠ dx =−62.4 ⎛ ⎝ 2 3 ⎞ ⎠ ⎡ ⎣ x 3 3 −1005 x 2 +253125 x ⎤ ≈ 5,015,230,000 lb = 2,507,615 t.
⎦ | 135 540
6.28 When the reservoir is at its average level, the surface of the water is about 50 ft below where it would be if the reservoir were full. What is the force on the face of the dam under these circumstances?
To learn more about Hoover Dam, see this article (http://www.openstax.org/l/20_HooverDam) published by the History Channel.
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
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