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Chapter 11: Problem 23

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{8}{x}, y=x^{2}, y=0, x=1, x=4 $$

Short Answer

Expert verified
The area of the region bounded by the graphs is \(8ln4 - \frac{64}{3} - 8ln1 + \frac{1}{3}\) square units.

Step by step solution

01

Sketch the Graphs

First, sketch the graphs of the functions. The function \(y = \frac{8}{x}\) is an inverse variation function that passes through points \((1, 8)\) and \((4, 2)\). The function \(y = x^{2}\) is a basic parabolic function which intersects with \(y = \frac{8}{x}\) at two points, \((2, 4)\) and \((4, 16)\). The lines \(y = 0\), \(x = 1\), and \(x = 4\) are the boundaries of the region.
02

Define Bounded Region

After sketching the graphs, identify the bounded region. That is the area enclosed between \(y=\frac{8}{x}\), \(y=x^{2}\) respect to the x-axis formed on the interval from \(x = 1\) to \(x = 4\). These functions represent the top and bottom curves that bound the region we are interested in.
03

Set up the Definite Integral

Set up the definite integral that will calculate the area. We subtract the y-values of the bottom function (\(y=x^2\)) from the top function (\(y=\frac{8}{x}\)) and integrate over the interval from 1 to 4. Accordingly, the integral becomes \(\int_1^4 (\frac{8}{x} - x^2) dx\).
04

Evaluate the Integral

To find the area, compute the integral. First, calculate the antiderivative of the functions: \(8ln|x| - \frac{x^3}{3}\). Then, using the Fundamental Theorem of Calculus, plug in the bounds of 1 and 4: \([8ln|4| - \frac{4^3}{3}]-[8ln|1| - \frac{1^3}{3}]\). Simplify this to find the area.

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Most popular questions from this chapter

The rate of change of mortgage debt outstanding for one- to four-family homes in the United States from 1998 through 2005 can be modeled by \(\frac{d M}{d t}=5.142 t^{2}-283,426.2 e^{-x}\) where \(M\) is the mortgage debt outstanding (in billions of dollars) and \(t\) is the year, with \(t=8\) corresponding to \(1998 .\) In 1998 , the mortgage debt outstanding in the United States was \(\$ 4259\) billion. (Source: Board of Governors of the Federal Reserve System) (a) Write a model for the debt as a function of \(t\). (b) What was the average mortgage debt outstanding for 1998 through \(2005 ?\)

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