We are often interested in knowing the area of a region. Forget momentarily that we addressed this already in Section 5. 5.4 and approach it instead using the technique described in Key Idea 7.0.1.

Let $Q$ be the area of a region bounded by continuous functions $f$ and $g\text{.}$ If we break the region into many subregions, we have an obvious equation:

Total Area = sum of the areas of the subregions.

The issue to address next is how to systematically break a region into subregions. A graph will help. Consider Figure 7.1.1 (a) where a region between two curves is shaded. While there are many ways to break this into subregions, one particularly efficient way is to “slice” it vertically, as shown in Figure 7.1.1 (b), into $n$ equally spaced slices.

We now approximate the area of a slice. Again, we have many options, but using a rectangle seems simplest. Picking any $x$-value $c_i$ in the $i^\text{ th }$ slice, we set the height of the rectangle to be $f(c_i)-g(c_i)\text{,}$ the difference of the corresponding $y$-values. The width of the rectangle is a small difference in $x$-values, which we represent with $\dx\text{.}$ Figure 7.1.1 (c) shows sample points $c_i$ chosen in each subinterval and appropriate rectangles drawn. (Each of these rectangles represents a differential element.) Each slice has an area approximately equal to $\big(f(c_i)-g(c_i)\big)\dx\text{;}$ hence, the total area is approximately the Riemann Sum \begin{equation*} Q = \sum_{i=1}^n \big(f(c_i)-g(c_i)\big)\dx. \end{equation*}

Taking the limit as $n\to \infty$ gives the exact area as $\int_a^b \big(f(x)-g(x)\big)\ dx\text{.}$

Example7.1.6Finding area enclosed by curves

Find the area of the region bounded by $f(x) = \sin(x) +2\text{,}$ $g(x) = \frac12\cos(2x)-1\text{,}$ $x=0$ and $x=4\pi\text{,}$ as shown in Figure 7.1.7.

Solution
Example7.1.8Finding total area enclosed by curves

Find the total area of the region enclosed by the functions $f(x) = -2x+5$ and $g(x) = x^3-7x^2+12x-3$ as shown in Figure 7.1.9.

Solution

The previous example makes note that we are expecting area to be positive. When first learning about the definite integral, we interpreted it as “signed area under the curve,” allowing for “negative area.” That doesn't apply here; area is to be positive.

The previous example also demonstrates that we often have to break a given region into subregions before applying Theorem 7.1.5. The following example shows another situation where this is applicable, along with an alternate view of applying the Theorem.

Example7.1.10Finding area: integrating with respect to $y$

Find the area of the region enclosed by the functions $y=\sqrt{x}+2\text{,}$ $y=-(x-1)^2+3$ and $y=2\text{,}$ as shown in Figure 7.1.11.

Solution

This calculus–based technique of finding area can be useful even with shapes that we normally think of as “easy.” Example 7.1.13 computes the area of a triangle. While the formula “$\frac12\times\text{ base } \times\text{ height }$” is well known, in arbitrary triangles it can be nontrivial to compute the height. Calculus makes the problem simple.

Example7.1.13Finding the area of a triangle

Compute the area of the regions bounded by the lines

$y=x+1\text{,}$ $y=-2x+7$ and $y=-\frac12x+\frac52\text{,}$ as shown in Figure 7.1.14.

Solution

While we have focused on producing exact answers, we are also able to make approximations using the principle of Theorem 7.1.5. The integrand in the theorem is a distance (“top minus bottom”); integrating this distance function gives an area. By taking discrete measurements of distance, we can approximate an area using numerical integration techniques developed in Section 5.5. The following example demonstrates this.

Example7.1.15Numerically approximating area

To approximate the area of a lake, shown in Figure 7.1.16 (a), the “length” of the lake is measured at 200-foot increments as shown in Figure 7.1.16 (b), where the lengths are given in hundreds of feet. Approximate the area of the lake.

Solution

In the next section we apply our applications–of–integration techniques to finding the volumes of certain solids.

Subsection7.1.1Exercises

Terms and Concepts

In the following exercises, find the area of the shaded region in the given graph.