Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5lb/ft^3.)

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Jick1984 · Accepted Answer

Circular pool has diameter 24 ft, so radius r=12 ft.  The sides are 5 ft high, water depth is 4 ft.  A horizotal &quot;slice&quot; of the weater with thickness △x&quot; has volume  V=πr2⋅△x=π(12)2△x=144π△x ft3  The force is obtained by multiplying by the weight per unit volume  F=144π△x ft3⋅62.5lbft3=9000π△x  If we let x = the heigh from the bottom of the pool, then each slice of water has to move a distance of  d=5−x  Because the side of the pool is 5 ft high.  If xi represents this height for the i th then we can write the Riemann sum  W=limn→∞∑i=0n9000πxi△x  To sum the total work, we integrate multiplying the force times distance. The slices become infinitely thin. The water is 4ft, so the limits are from 0 to 4.  W=∫04(9000π)(5−x)dx  =9000π[5x−12x2]04  =900π[20−8−(0)]  =108000π≈339292 ft-lb

Show how to approximate the required work by a Riemann sum. Then expre

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