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.

First, let's sketch the region and the axis of rotation to get a better visual understanding of the problem. The given region is a bounded area between the curves y = e^(2x) + 1 and y = 0 for the x values between 0 and 0.9.

Next, we need to find the expression for the volume generated by a single cylindrical shell. The volume of a cylindrical shell is given by the formula:

V = 2πrh * Δx

Where:
- V is the volume of the shell
- π is the mathematical constant pi (approximately 3.14159)
- r is the distance of the shell from the axis of rotation (in this case, the y-axis)
- h is the height of the shell
- Δx is the thickness of the shell (in this case, the difference between x and x+Δx)

In our case, to express the variables r and h in terms of x, we need to rewrite the equations of the curves in terms of x.

For y = e^(2x) + 1, let's solve for x:

y = e^(2x) + 1
e^(2x) = y - 1
2x = ln(y - 1)
x = (1/2) * ln(y - 1)

Now we have x in terms of y. Let's find the height h:

h = y

Since the region is being rotated about the y-axis, the distance r is simply the x-coordinate for each shell, which is equal to x.

Now we can calculate the volume of a single shell:

V = 2πrh * Δx
V = 2π(x)(y) * Δx

To find the total volume, we integrate this expression over the given range of x from 0 to 0.9:

V = ∫[0 to 0.9] 2π(x)(y) dx

Now, to evaluate this integral, we substitute the expression for y:

V = ∫[0 to 0.9] 2π(x)((e^(2x) + 1)) dx

Finally, we perform the integral numerically using numerical methods or solve it symbolically using mathematical software to find the exact volume.

Note: The exact numerical value of the volume will depend on the specific values of x and the accuracy of the calculation method used.