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 given curves about the y-axis, we can use the method of cylindrical shells.

First, let's draw a graph of the given curves to understand the region we are working with.

The first curve is y = e^(2x) + 1. This is an exponential curve that starts at y = 1 when x = 0 and increases rapidly as x increases.

The second curve is y = 0. This is just the x-axis, which acts as the lower bound of the region.

The third curve is x = 0. This is the y-axis, which acts as the left bound of the region.

The fourth curve is x = 0.9. This is a vertical line that intersects the exponential curve at one point.

Now, we can visualize the region enclosed by these curves.

To find the volume, we'll use the formula for the volume of a solid of revolution about the y-axis:

V = ∫[a,b] 2πx * h(x) * dx

Where a and b are the limits of integration, and h(x) represents the height of the cylindrical shell at each value of x.

In this case, the limits of integration are from y = 0 to y = e^(2 × 0.9) + 1. Since we're rotating about the y-axis, we need to express everything in terms of y.

Let's rewrite the equations in terms of y:

For the exponential curve, y = e^(2x) + 1
=> e^(2x) = y - 1
=> 2x = ln(y - 1)
=> x = (ln(y - 1))/2

Now, let's rewrite the volume formula:

V = ∫[0,e^(2 × 0.9) + 1] 2πx * h(x) * dx
= ∫[0,e^(2 × 0.9) + 1] 2π((ln(y - 1))/2) * (y - 0) * dx

Simplifying further:

V = π∫[0,e^(2 × 0.9) + 1] (ln(y - 1)) * (y - 0) * dx

Now, we need to evaluate this integral to find the volume.