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 532.]
Enclosed by y = ex, y = 2x + 1, x = −3, and x = 1
I suspect you mean y = e^x
confirm before I attempt this.
To find the area of the indicated region enclosed by the given curves, we can use the definite integral.
First, let's find the points where the curves intersect. Setting the equations equal to each other, we have:
ex = 2x + 1
To solve this equation, we can use numerical methods, such as graphing or using technology. By observing the graph or using a graphing calculator, we can see that the curves intersect approximately at x ≈ -1.678.
Now, we need to evaluate the definite integral to find the area between the curves. The area can be calculated as follows:
∫[-3, -1.678] (2x + 1 - ex) dx + ∫[-1.678, 1] (ex - 2x - 1) dx
To evaluate these integrals, we can either use technology or antidifferentiate each term separately:
∫[-3, -1.678] (2x + 1 - ex) dx = [x^2 + x - ex] |[-3,-1.678]
∫[-1.678, 1] (ex - 2x - 1) dx = [ex - x^2 - x] |[-1.678, 1]
Evaluating these integrals at their respective limits and using the values we obtained from the intersections earlier, we can compute the area:
([1^2 + 1 - e^1] - [(-3)^2 + (-3) - e^(-3)]) + (e^1 - 1^2 - 1) - (e^(-1.678) - (-1.678)^2 - (-1.678))
After performing the arithmetic, the resulting value will give us the area of the region enclosed by the given curves. Remember to round the answer to four significant digits.