Question

Determine the area of the region enclosed by the curves y = e^x, y = xe^x, and x = 0.

a) 1/2
b) 1/e
c) e - 1
d) e + 1

Answer 1

To find the area of the region enclosed by the curves y = e^x, y = xe^x, and x = 0, we need to find the points of intersection between these curves and evaluate the integral of the difference between the curves.

Step-by-step explanation:

To find the area of the region enclosed by the curves y = e^x, y = xe^x, and x = 0, we need to find the points of intersection between these curves.

Setting y = e^x and y = xe^x equal to each other, we have e^x = xe^x. Dividing both sides by e^x, we get x = 1.

The area of the region is given by the integral of the difference between the curves between the x-values 0 and 1: A = ∫(e^x - xe^x)dx from 0 to 1. Evaluating this integral gives us the area of the region enclosed by the curves.