Problem 38 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.