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Chapter 11: Problem 38

Use a graphing utility to graph the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=-x^{2}+4 x+2, g(x)=x+2 $$

Short Answer

Expert verified
The steps to find the area of the region between the given functions include graphing the functions \(f(x)=-x^{2}+4 x+2\) and \(g(x)=x+2\), finding their points of intersection, and then computing the definite integral of the absolute difference of the two functions over the interval between the points of intersection. The numerical result of the integral calculation represents the area of the region.

Step by step solution

01

Graph the Functions

To visualize the region in question, start by graphing both functions \(f(x)=-x^{2}+4 x+2\) and \(g(x)=x+2\). By doing so, the area between these two functions can be clearly seen.
02

Find Points of Intersection

Next, find the points where \(f(x)\) and \(g(x)\) intersect. Set these two equations equal to each other: \(-x^{2}+4 x+2=x+2\) and then solve for \(x\). This is where two functions intersect, and the result gives the boundaries for the integration in the next step.
03

Calculate the Area

The area between two curves \(f(x)\) and \(g(x)\) from \(a\) to \(b\) is given by the integral of the absolute difference of the two functions over the interval \([a, b]\), as presented in this formula: \[Area = \int_{a}^{b}|f(x) - g(x)| dx\]. Substitute \(f(x)\), \(g(x)\), \(a\), and \(b\) into the formula with your values and compute the definite integral.

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Most popular questions from this chapter

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{1}{2} x\right] d x $$

Use a graphing utility to graph the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=3-2 x-x^{2}, g(x)=0 $$

Use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all \(x\) -values in the interval for which the function is equal to its average value. $$ f(x)=\frac{1}{(x-3)^{2}} \quad[0,2] $$

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Find the area of the region. $$ \begin{aligned} &f(x)=3\left(x^{3}-x\right) \\ &g(x)=0 \end{aligned} $$
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