The base of a solid is the region in the first quadrant bounded by the y-axis, the x-axis, the graph of y=e^x, and the vertical line x=1. For this solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid
To find the volume of the solid, we need to integrate the area of each square cross section along the x-axis from x=0 to x=1.
The side length of each square is given by y = e^x, since the height of each square is determined by the function e^x. Therefore, the area of each square is (e^x)^2 = e^(2x).
We can integrate this area function from x=0 to x=1:
∫[0,1] e^(2x) dx = [1/2 * e^(2x)] [0,1] = 1/2 * (e^2 - e^0) = 1/2 * (e^2 - 1)
Therefore, the volume of the solid is 1/2 * (e^2 - 1) cubic units.