Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer. (Round your answer to four significant digits.)
Enclosed by y = e^x, y = 2x + 1, x = −2, and x = 1
To find the area of the region enclosed by the given curves, we first need to determine the points where the curves intersect.
To do this, let's set the two given equations equal to each other:
e^x = 2x + 1
This equation is a transcendental equation, meaning it cannot be solved algebraically. We need to use numerical methods or technology to find the approximate solutions.
Let's use a graphing calculator or any other graphing tool to graph the curves y = e^x and y = 2x + 1. By inspecting the graph or using an intersection feature, we can find the x-values where the curves intersect.
Once we have determined the x-values of the intersection points, we can integrate the difference between the two curves over the defined x-interval, which is from x = -2 to x = 1.
Let's set up the integral:
∫[a to b] [(top function) - (bottom function)] dx
In this case, the top function is y = e^x, and the bottom function is y = 2x + 1. The interval of integration is from x = -2 to x = 1, which we found from the graph.
∫[-2 to 1] [e^x - (2x + 1)] dx
Now we can evaluate this integral using numerical methods or technology, such as a graphing calculator or symbolic computation software.
Once we have the result of the integral, we can round the answer to four significant digits as required and obtain the area of the indicated region.
look at the graph first.
http://www.wolframalpha.com/input/?i=plot+y+%3D+e%5Ex%2C+y+%3D+2x%2B1%2C+-2+to+1
I is obvious that y = e^x and y = 2x+1 intersect at the y-axis at (0,1). there is another intersection but it beyond x = 1, so of no concern to us
So area
= ∫ (e^x - 2x - 1)dx from -2 to 0 + ∫(2x + 1 - e^x) dx from 0 to 1
the integration and evaluation are straightforward