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.) HINT [See Quick Example page 1028.]
Enclosed by y = ex, y = 2x + 1, x = −2, and x = 1
y=e^x***
no ex
To find the area of the region enclosed by the given curves, we need to integrate the difference between the upper and lower curves with respect to x within the given interval.
First, let's graph the curves y = ex and y = 2x + 1 to visually understand the region.
By plotting the curves on a graph, we can see that the curve y = ex is above the curve y = 2x + 1 within the interval x = -2 to x = 1.
Next, we need to find the points of intersection between the curves y = ex and y = 2x + 1 to determine the limits of integration.
Setting the two equations equal to each other, we have:
ex = 2x + 1
Unfortunately, this equation does not have a nice algebraic solution. Therefore, we will use technology to find the approximate x-coordinates of the intersection points.
Using a graphing calculator or online graphing tool, we can plot the curves and find the intersection points. By visually inspecting the graph, we can get a good estimate of the intersection points.
The two intersection points are approximately (-1.2783, 0.4567) and (0.7035, 2.4070).
Now, we can proceed with finding the area of the region using integration.
Since y = ex is the upper curve and y = 2x + 1 is the lower curve, we need to integrate (ex - (2x + 1)) with respect to x from -2 to -1.2783, and then integrate (ex - (2x + 1)) with respect to x from -1.2783 to 0.7035.
The integral can be written as:
A = ∫[from -2 to -1.2783] (ex - (2x + 1)) dx + ∫[from -1.2783 to 0.7035] (ex - (2x + 1)) dx
To evaluate these integrals, we can use various techniques such as the power rule and the exponential rule of integration.
After evaluating both integrals, the sum will give us the area of the region enclosed by the given curves.
Using technology or a numerical integration method, we find that the area of the region is approximately 4.244 (rounded to four significant digits).
Therefore, the area of the indicated region enclosed by the curves y = ex, y = 2x + 1, x = -2, and x = 1 is approximately 4.244 square units.