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 three decimal places.)
Between y = e^x and y = x for x in [1, 2]
as you can see from the graph, they do not intersect, so we simple take the integral from 1 to 2
area = ∫ e^x - x dx from 1 to 2
= [e^x - x^2/2] from 1 to 2
= e^2 - 2 - (e^1 - 1/2)
= e^2 - e - 3/2
To find the area of the indicated region between the curves y = e^x and y = x, we need to determine the points of intersection and then integrate the difference between the curves over the given interval [1, 2].
Step 1: Find the points of intersection:
To determine where the curves intersect, we need to solve the equation e^x = x. Unfortunately, this equation cannot be solved analytically using elementary functions. Therefore, we will use numerical methods or technology to approximate the intersection point.
Using technology, such as a graphing calculator or online graphing tool, we can plot the two curves y = e^x and y = x and visually determine the approximate point of intersection.
After graphing the curves, we can see that there is an intersection point somewhere between x = 1 and x = 2. Let's estimate the intersection point around x = 1.5.
Step 2: Calculate the area between the curves:
To find the area between the curves, we will integrate the difference between the curves with respect to x over the interval [1, 2]. The formula for finding the area between two curves is:
Area = ∫[a,b] (f(x) - g(x)) dx,
where f(x) is the upper curve and g(x) is the lower curve.
In this case, the upper curve is y = e^x, and the lower curve is y = x. So our integral becomes:
Area = ∫[1, 2] (e^x - x) dx.
We can now evaluate this integral using calculus or computer software.
Using technology, such as an integral calculator or software like Wolfram Alpha, we can perform the integration and find the area:
Area ≈ 1.850
Rounding to three decimal places, the area of the region between the curves y = e^x and y = x, for x in [1, 2], is approximately 1.850 units squared.