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

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 fit, the sides are $5 \mathrm{ft}$ high, and the depth of the water is $4 \mathrm{ft}$. How much work is required to purnp all of the water out over the side? (Use the fact that water weighs $62.5 \mathrm{lb} / \mathrm{ft}^{3} .$ )

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 fit, the sides are
$5 \mathrm{ft}$ high, and the depth of the water is $4 \mathrm{ft}$. How much work is required to purnp all of the water out over the side? (Use the fact that water weighs $62.5 \mathrm{lb} / \mathrm{ft}^{3} .$ )

Key Concepts

Density and Volume Calculations

Density is a measure of mass (or weight) per unit volume, and it is used to convert the volume of water into weight. In the work problem, the density of water is used to calculate the force required to lift each small slice. Knowing the geometric dimensions of the pool and the thickness of each slice allows for an expression of the volume, which, when multiplied by the density, gives the weight, and hence the force for the work computation.

Work in Physics

Work in physics is defined as the product of force and displacement. For problems involving fluid pumping, the work is computed by considering the weight (force) of a small volume of water and the vertical distance this water must be lifted. The overall work is then found by integrating these contributions over the depth of the water.

Slicing Method

The slicing method involves cutting the region (or the body of water) into small horizontal layers or slices. For each slice, the individual work needed is determined based on its position and the distance it must be raised. This method allows one to efficiently set up the integral by relating the volume, density, and lift height of each thin slice of water.

Riemann Sum

A Riemann sum is a method for approximating the value of an integral by summing the areas (or contributions) of many small subintervals or slices. In the context of the work problem, the water is divided into thin layers, and the work to pump each layer is computed separately; the sum of these contributions approximates the overall work needed to pump out the water.

Definite Integral

A definite integral is the limit of a Riemann sum as the number of slices approaches infinity, providing an exact calculation of the total accumulated quantity—in this case, the total work. It is used to sum up infinitesimally small contributions over a continuous interval, which makes it ideal for modeling how work is done to move each differential volume of water over a given distance.

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