Question

Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ y=\frac{4}{x}, y=x, x=1, x=4 $$

Short answer

The definite integrals that represent the area of the region are \(\int_1^2 (\frac{4}{x} - x)\,dx\) and \(\int_2^4 (x-\frac{4}{x})\,dx\).

Step-by-step solution

Understanding the Functions

Start by looking at the functions individually. The function \(y=\frac{4}{x}\) is a hyperbola with the center at origin. The function \(y=x\) is a straight line passing through the origin with a slope of 1. The other two conditions \(x=1\) and \(x=4\) are vertical lines at \(x=1\) and \(x=4\) respectively.

Graphing the Functions

Using a graphing utility, plot the graphs of each of these functions on the same xy-plane. The region of interest is bounded by the graphs of the functions \(y=\frac{4}{x}\), \(y=x\), \(x=1\), and \(x=4\). Notice that the two curves intersect at points (1,1) and (4,4) and hence forms a closed region.

Setting up the Integrals

This region can be divided into two parts. One from \(x=1\) to \(x=2\) where \(y=\frac{4}{x}\) is above \(y=x\) and the other from \(x=2\) to \(x=4\) where \(y=x\) is above \(y=\frac{4}{x}\). The area of the region can be calculated by setting up two integrals, one from \(x=1\) to \(x=2\) where the integrand is \(\frac{4}{x} - x\), and the other from \(x=2\) to \(x=4\) where the integrand is \(x-\frac{4}{x}\).

Computing the Area

Now, compute the definite integrals. The definite integrals that represent the area of the region are \(\int_1^2 (\frac{4}{x} - x)\,dx\) and \(\int_2^4 (x-\frac{4}{x})\,dx\). Solving these integrals will give the area of the region.