694
Chapter 6 | Applications of Integration
From properties of similar triangles, we have
r i 12− x i * =
4 12 = 1 3 3 r i = 12− x i * r i = 12− x i * 3 = 4− x i * 3 .
Then the volume of the disk is
⎛ ⎝ 4−
⎞ ⎠
2
x i * 3
V i = π Δ x (step 2). The weight-density of water is 62.4 lb/ft 3 , so the force needed to lift each layer is approximately
⎛ ⎝ 4−
⎞ ⎠
2
x i * 3
F i ≈62.4 π
Δ x (step 3).
Based on the diagram, the distance the water must be lifted is approximately x i * feet (step 4), so the approximate work needed to lift the layer is
⎛ ⎝ 4−
⎞ ⎠
2
x i * 3
W i ≈62.4 πx i *
Δ x (step 5).
Summing the work required to lift all the layers, we get an approximate value of the total work: W = ∑ i =1 n W i ≈ ∑ i =1 n 62.4 πx i * ⎛ ⎝ 4− x i * 3 ⎞ ⎠ 2 Δ x (step 6).
Taking the limit as n →∞, we obtain
⎛ ⎝ 4−
⎞ ⎠
2
→∞ ∑ i =1 n
x i * 3
W = lim n
62.4 πx i *
Δ x
8
⎛ ⎝ 4− x 3
⎞ ⎠
2
= ∫
62.4 πx
dx
0
8 ⎛ ⎝ 16 x − 8 x
⎞ ⎠ dx
x ⎛
⎞ ⎠ dx =62.4 π ∫ 0
8
x 3 9
2 3 +
x 2 9
=62.4 π ∫
⎝ 16− 8 x
3 +
0
⎤ ⎦ | 0 8
⎡ ⎣ 8 x 2 − 8 x
3 9 +
x 4 36
=62.4 π =10,649.6 π ≈ 33,456.7. It takes approximately 33,450 ft-lb of work to empty the tank to the desired level.
6.26 A tank is in the shape of an inverted cone, with height 10 ft and base radius 6 ft. The tank is filled to a depth of 8 ft to start with, and water is pumped over the upper edge of the tank until 3 ft of water remain in the tank. How much work is required to pump out that amount of water?
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