The region enclosed by the graph e^(x/2), y=1, and x=ln(3) is revolved around the x-axis. Find the volume of the solid generated.

I don't understand if we have to use the washer method or the disk method for this one because when I drew it out on a graph it looked very confusing.

what does y = e^.5 x look like?

at x = 0, y =e^0 = 1
at x = .1, y .= e^.05 = 1.05
at x = .2 y = 1.11

at x = ln(3) = 1.098 = 1.1
y = e^.5 ln (3) = e^ln (sqrt 3) = 1.73

well, y goes from 1 at x = 0
to
y = 1.73 at x =1.1l
so it looks like washers
but the sensible way might be to integrate
y = pi (e^.5x)^2dx from x = 0 to x
= 1.1
and
from that subtract the cylinder
pi (1)^2 dx from 0 to 1.1

As above, using discs of thickness dx,

v = ∫[0,ln3] π(R^2-r^2) dx
where R=y and r=1
v = ∫[0,ln3] π((e^(x/2)^2-1^2) dx
= ∫[0,ln3] π(e^x-1) dx
= π(2-ln3)

or, using shells of thickness dy,

v = ∫[1,√3] 2πrh dy
where r=y and h=ln3-x
v = ∫[1,√3] 2πy(ln3-2lny) dy
= π(2-ln3)

To find the volume of the solid generated when the region enclosed by the graph of the function e^(x/2), y=1, and x=ln(3) is revolved around the x-axis, we need to use the disk method.

First, let's understand the region that needs to be revolved. The graph of the function e^(x/2) is an increasing exponential curve that starts from (0, 1). The line y=1 is a horizontal line passing through y=1. The vertical line x=ln(3) intersects the graph of the function at a certain point. The region enclosed by these three curves is a finite shape.

Now, let's break down the process to find the volume using the disk method:

1. Determine the limits of integration:
In this case, the limits of integration will be from 0 to ln(3), as these are the x-values where the region enclosed by the curves begins and ends.

2. Set up the integral:
The volume using the disk method is given by the formula:
V = ∫[a, b] π * [f(x)]^2 dx

Here, a = 0 and b = ln(3). f(x) represents the distance between the curve e^(x/2) and the axis of rotation (the x-axis).

3. Express f(x) in terms of x:
In this case, f(x) will be the function e^(x/2) since it represents the distance between the curve and the x-axis.

4. Square f(x):
Squaring the function f(x), we get [e^(x/2)]^2 = e^x.

5. Calculate the definite integral:
V = ∫[0, ln(3)] π * e^x dx

6. Evaluate the integral:
Integrate the function π * e^x with respect to x from 0 to ln(3) to get the volume.

By following these steps and evaluating the integral, you will find the volume of the solid generated by revolving the region around the x-axis.