The Area Problem and Integral Calculus

We now turn our attention to a classic question from calculus. Many quantities in physics—for example, quantities of work—may be interpreted as the area under a curve. This leads us to ask the question: How can we find the area between the graph of a function and the x-axis over an interval (Figure 2.8)?

A graph is shown of a generic curved function f(x) shaped like a hill in quadrant one. An area under the function is shaded above the x-axis and between x=a and x=b.

Figure 2.8 The Area Problem: How do we find the area of the shaded region?

As in the answer to our previous questions on velocity, we first try to approximate the solution. We approximate the area by dividing up the interval \(\left[\right. a , b \left]\right.\) into smaller intervals in the shape of rectangles. The approximation of the area comes from adding up the areas of these rectangles (Figure 2.9).

The graph is the same as the previous image, with one difference. Instead of the area completely shaded under the curved function, the interval [a, b] is divided into smaller intervals in the shape of rectangles. The rectangles have the same small width. The height of each rectangle is the height of the function at the midpoint of the base of that specific rectangle.

Figure 2.9 The area of the region under the curve is approximated by summing the areas of thin rectangles.

As the widths of the rectangles become smaller (approach zero), the sums of the areas of the rectangles approach the area between the graph of \(f \left(\right. x \left.\right)\) and the x-axis over the interval \(\left[\right. a , b \left]\right. .\) Once again, we find ourselves taking a limit. Limits of this type serve as a basis for the definition of the definite integral. Integral calculus is the study of integrals and their applications.

Example 2.3

Estimation Using Rectangles

Estimate the area between the x-axis and the graph of \(f \left(\right. x \left.\right) = x^{2} + 1\) over the interval \(\left[\right. 0 , 3 \left]\right.\) by using the three rectangles shown in Figure 2.10.

A graph of the parabola f(x) – x^2 + 1 drawn on graph paper with all units shown. The rectangles completely contained under the function and above the x-axis in the interval [0,3] are shaded. This strategy sets the heights of the rectangles as the smaller of the two corners that could intersect with the function. As such, the rectangles are shorter than the height of the function.

Figure 2.10 The area of the region under the curve of \(f \left(\right. x \left.\right) = x^{2} + 1\) can be estimated using rectangles.

Solution

The areas of the three rectangles are 1 unit², 2 unit², and 5 unit². Using these rectangles, our area estimate is 8 unit².

Checkpoint 2.3

Estimate the area between the x-axis and the graph of \(f \left(\right. x \left.\right) = x^{2} + 1\) over the interval \(\left[\right. 0 , 3 \left]\right.\) by using the three rectangles shown here:

A graph of the same parabola f(x) = x^2 + 1, but with a different shading strategy over the interval [0,3]. This time, the shaded rectangles are given the height of the taller corner that could intersect with the function. As such, the rectangles go higher than the height of the function.

This lesson is part of:

Limits

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