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We apply the procedure of “Slice, Approximate, Integrate” to model physical situations.

Thus far, we have studied several geometric applications of the procedure “Slice, Approximate, Integrate.” Indeed, it can be used to find lengths, areas, volumes. This procedure is not limited to modeling only geometric situations. Many problems from other STEM fields requires the same technique. This section examines many of these situations. First, we make a more generalized observation of the philosophy behind the “Slice, Approximate, Integrate” procedure.

Let’s take a step back and try to think about each of the situations we’ve used the “Slice, Approximate, Integrate” procedure to model. For each quantity of interest - length, area, or volume- we used an object whose length, area, or volume we could calculate.

• To find the area between curves, we know how to find the area of rectangles, so we approximate each slice by a rectangle.
• For solids of revolution, we can calculate the volume of washers or shell, so we approximate each slice by a washer or a shell.
• For length of curves, we can find the length of a line segment, so we approximate each slice by a line segment.

In physics, we take measurable quantities from the real world and attempt to find meaningful relationships between them. These relationships are often expressed using formulas, but these come with assumptions or restrictions. If these requirements are not met, we can still apply the “Slice, Approximate, Integrate” procedure to compute physical quantities of interest.

### Mass of a wire with variable density

Given a physical object, its density is a measure of how the mass of the object is distributed. In the case where the density of the object is constant, we have the following.

Three dimensions: $\left <\textrm {density}\right > = \frac {\left <\textrm {mass}\right >}{\left <\textrm {volume}\right >}$

Two dimensions: $\left <\textrm {density}\right > = \frac {\left <\textrm {mass}\right >}{\left <\textrm {area}\right >}$

One dimension: $\left <\textrm {density}\right > = \frac {\left <\textrm {mass}\right >}{\left <\textrm {length}\right >}$

We can (and often do) approximate physical objects, like wires, as one dimensional or thin sheets as two dimensional.

If a wire has linear density $5 \unit {g}/\unit {m}$, how many grams will $4 \unit {m}$ of this wire be?
In this case, we need just multiply the density by the length.

Sometimes the linear density of an object can vary from one part of the object to another. In this case, density will be a non-constant function, which we will represent by the Greek letter, rho, $\rho$ (pronounced “rho”). In this case, the above formula no longer allows us to compute the mass of the wire. Let’s tackle this scenario with a motivating example.

There was nothing particularly special about this example. In fact, we can summarize the results in a formula.

### Work

One of the most important concepts in physics is that of work, which measures the change in energy that occurs when a force moves an object over a certain displacement. For those familiar with physics, the Work-Energy Theorem is a powerful tool for studying situations in Newtonian mechanics (such as a box sliding down an incline plane).

For a constant force $F$ acting on a particle over a displacement $d$ in the direction of displacement, work is given by the formula:

Denoting work by $W$, force by $F$, and displacement by $d$, we can write:

In what follows, we will look at two scenarios in which the formula $W=F \cdot d$ does not immediately apply. They result when either the force required to move an object is not constant, or when different parts of the object we move must be moved different distances. They can be summarized by:

#### Work done by a non-constant force on a particle

What will happen if the work done over the displacement is not constant? We study another motivating example that leads to a more general result.

Let’s try some examples to see the formula in action.

#### Work done by a constant force on a collection of particles

In the last problems, a non-constant force acted on a single particle. Another scenario occurs when a constant force acts on a collection of particles. The following example explores this situation in detail.

As usual, the process here can be generalized into another formula.

A common instance in these problems requires that the liquid be moved above the vertical distance to which the tank is filled. By using the conventions above, the mathematical condition for this is expressed by requiring that $h$h>a$$h$h>b$ , and with this condition, $D(y)=$ $b-h$$h-b$$b-y$$y-b$$h-y$$y-h$ .

### Final thoughts

The technique of “Slice, Approximate, Integrate” can be used to solve physical problems as well as geometric ones. These are not the only examples of physical problems that can be modeled and solved by using this technique. The exercises will explore other problems, and you will run into many more in other STEM courses.

“Mathematics is the abstract key which turns the lock of the physical universe” - John Polkinghome