Areas Between Curves: Learn It 2

Areas of Compound Regions

So far, we have required [latex]f(x)\ge g(x)[/latex] over the entire interval of interest, but what if we want to look at regions bounded by the graphs of functions that cross one another? In that case, we modify the process we just developed by using the absolute value function.

finding the area of a region between curves that cross

Let [latex]f(x)[/latex] and [latex]g(x)[/latex] be continuous functions over an interval [latex]\left[a,b\right].[/latex] Let [latex]R[/latex] denote the region between the graphs of [latex]f(x)[/latex] and [latex]g(x),[/latex] and be bounded on the left and right by the lines [latex]x=a[/latex] and [latex]x=b,[/latex] respectively. Then, the area of [latex]R[/latex] is given by:

[latex]A={\displaystyle\int }_{a}^{b}|f(x)-g(x)|dx[/latex]

In practice, applying this theorem requires us to break up the interval [latex]\left[a,b\right][/latex] and evaluate several integrals, depending on which of the function values is greater over a given part of the interval. We study this process in the following example.

If [latex]R[/latex] is the region between the graphs of the functions [latex]f(x)= \sin x[/latex] and [latex]g(x)= \cos x[/latex] over the interval [latex]\left[0,\pi \right],[/latex] find the area of region [latex]R.[/latex]

Consider the region depicted in the following figure. Find the area of [latex]R.[/latex]

This figure is has two graphs in the first quadrant. They are the functions f(x) = squareroot of x and g(x)= 3/2 – x/2. In between these graphs is a shaded region, bounded to the left by f(x) and to the right by g(x). All of which is above the x-axis. The shaded area is between x=0 and x=3.
Figure 7.