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

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
Circular pool has diameter 24 ft, so radius r=12 ft.
The sides are 5 ft high, water depth is 4 ft.
A horizotal "slice" of the weater with thickness $\mathrm{△}x$" has volume

The force is obtained by multiplying by the weight per unit volume

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 ${x}_{i}$ represents this height for the i th then we can write the Riemann sum
$W=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=0}^{n}9000\pi {x}_{i}\mathrm{△}x$
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.
$W={\int }_{0}^{4}\left(9000\pi \right)\left(5-x\right)dx$
$=9000\pi {\left[5x-\frac{1}{2}{x}^{2}\right]}_{0}^{4}$
$=900\pi \left[20-8-\left(0\right)\right]$
$=108000\pi \approx 339292$ ft-lb