13-20 Show how to approximate the required work by a Riemann...

$13-20$ Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. $$\begin{array}{l}{\text { A circular swimming pool has a diameter of } 24 \mathrm{ft} \text { , the sides }} \\ {\text { are } 5 \mathrm{ft} \text { high, and the depth of the water is } 4 \mathrm{ft} \text { . How much }} \\ {\text { work is required to pump all of the water out over the side? }} \\ {\text { (Use the fact that water weighs } 62.5 \mathrm{lb} / \mathrm{ft}^{3} . \text { ) }}\end{array}$$

$13-20$ Show how to approximate the required work by a Riemann
sum. Then express the work as an integral and evaluate it.
$$\begin{array}{l}{\text { A circular swimming pool has a diameter of } 24 \mathrm{ft} \text { , the sides }} \\ {\text { are } 5 \mathrm{ft} \text { high, and the depth of the water is } 4 \mathrm{ft} \text { . How much }} \\ {\text { work is required to pump all of the water out over the side? }} \\ {\text { (Use the fact that water weighs } 62.5 \mathrm{lb} / \mathrm{ft}^{3} . \text { ) }}\end{array}$$

Key Concepts

Weight Density

Weight density is a measure of weight per unit volume and is fundamental when dealing with fluids in work calculations. In this context, the weight density multiplies the volume of each slice to yield the weight, which is then used to determine the force needed to move that slice. This concept connects the physical properties of a material with the mathematical formulation of work through integration.

Slicing Method

The slicing method involves breaking down a physical system into small, manageable pieces or slices where the physical quantities involved (like force or weight) can be considered approximately constant. In problems where work is computed, the method involves evaluating the work done on each slice and then integrating over the entire domain to capture the cumulative effect.

Riemann Sum Approximation

A Riemann sum is a method for approximating the total value of a function over an interval by dividing the area (or other quantities) into a finite number of slices or subintervals. In the context of computing work, this approach involves partitioning the volume of water into thin slices and summing the work required to move each slice, which sets the foundation for transitioning to an integral representation as the slice thickness approaches zero.

Definite Integral

A definite integral represents the limit of a Riemann sum as the number of subintervals increases indefinitely while their width decreases. It provides an exact value for the total accumulation of a quantity, such as work, over a continuous interval. In work problems, the definite integral integrates force over a distance, accounting for variations in the force as a function of position.

Work (Integration in Physics)

Work is defined as the product of force and displacement. When the force is variable, such as in the case of lifting water, the work done is best computed using integration. This concept links the physical notion of work with the mathematical tool of the integral, allowing one to sum an infinite number of infinitesimally small contributions to obtain the total work.

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