The volume of the solid obtained by rotating the region bounded by y=e^x, y=Inx, x=1, and x=2 about the line y-axis can be computed using the method of cylindrical shells. Using the method of cylindrical shells find the volume.
Draw the curves.
v = ∫[1,2] 2πrh dx
where r=x and h=e^x-ln(x)
v = ∫[1,2] 2πx(e^x-lnx) dx = 2π(e^2-ln4)
using discs requires breaking up the volume at boundary changes, and that's too much bother for now. It would be
∫[0,ln2] π((e^y)^2-1^2) dy
+ ∫[ln2,e] π(2^2-1^2) dy
+ ∫[e,e^2] π(2^2-(lny)^2) dy
check my setup
Why did the region bounded by y=e^x, y=Inx, x=1, and x=2 decide to join a circus?
Because it wanted to become a "rotating" act! Ba dum tss!
Alright, let's get serious now. To find the volume using the method of cylindrical shells, we'll integrate the circumference of each cylindrical shell multipled by its height over the range of x from 1 to 2.
The shell's height is given by deltaY = e^x - ln(x) and its radius is x. So the circumference is 2πx and the volume of each shell is (2πx)(e^x - ln(x)).
Integrating the volume of each shell from x=1 to x=2 gives us the total volume of the region:
V = ∫[1 to 2] (2πx)(e^x - ln(x)) dx
Now that's a math problem that really "shells" out the laughs, isn't it? Let's compute the integral and find the volume!
To find the volume using the method of cylindrical shells, we first need to set up the integral based on the given region and axis of rotation.
The region bounded by y=e^x, y=ln(x), x=1, and x=2 can be visualized as shown below:
```
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x=1 x=2
```
To use the cylindrical shell method, we need to consider the cross-sections perpendicular to the axis of rotation (in this case, the y-axis). Each cylindrical shell will have a radius equal to the x-coordinate and a width given by the difference in y-values between the curves e^x and ln(x).
The volume of each cylindrical shell can be calculated using the formula for the volume of a cylinder, which is V = 2πrh * Δh, where r is the radius, h is the height (or width), and Δh is the thickness of each shell.
In this case, the radius r is given by the x-coordinate, h is the difference between y=e^x and y=ln(x), and Δh is the infinitesimal thickness (dh) of each shell.
Now, let's set up the integral to find the volume using the cylindrical shell method:
V = ∫[a, b] 2πx (e^x - ln(x)) dx,
where [a, b] represents the interval of integration based on the given x-values (in this case, a=1 and b=2).
To evaluate this integral, we need to find the antiderivative of (e^x - ln(x)) with respect to x, which involves integrating each term separately.
After evaluating the integral, the resulting expression will give us the volume of the solid obtained by rotating the region bounded by y=e^x, y=ln(x), x=1, and x=2 about the y-axis using the method of cylindrical shells.