Find the volume of the solid formed by rotating the region enclosed by

y=e^(2x)+1 y=0 x=0 x=0.9
about the y-axis.

To find the volume of the solid formed by rotating the region enclosed by the curves around the y-axis, we can use the method of cylindrical shells.

The first step is to determine the limits of integration. In this case, we are given the equations y = e^(2x) + 1 and y = 0, and we need to find the x-values when these curves intersect. Solving for x, we have:

0 = e^(2x) + 1
e^(2x) = -1 (No real solutions)

Since the curves do not intersect, the region enclosed is formed by y = e^(2x) + 1 and the x-axis. The limits of integration are x = 0 and x = 0.9.

Next, we determine the height of each cylindrical shell. At any given x-value, the height of the shell is given by the difference between the curve and the x-axis. In this case, it is y = e^(2x) + 1.

The radius of each shell is the x-value. So, the radius is equal to x.

The differential volume of each shell is given by 2πrh*dx, where r is the radius, h is the height, and dx is the thickness of the shell.

To find the volume, we integrate this expression from x = 0 to x = 0.9:

V = ∫(0 to 0.9) 2πx(e^(2x) + 1) dx

Evaluating this integral will give you the volume of the solid formed when the region is rotated about the y-axis.