Consider the region R bounded by f(x) = ln x, x = e, and the x-axis. The units on the axes are in meters.
Set up, but do not evaluate, an integral representing the volume of the solid when R is rotated
about the x-axis. Include units.
To find the volume of the solid when region R is rotated about the x-axis, we can use the method of cylindrical shells.
Cylindrical shells are like the layers of an onion. Each shell has a certain height, circumference, and thickness. When we add up the volumes of all these shells, we get the total volume of the solid.
First, let's visualize the region R and the shell. R is bounded by the graph of f(x) = ln x, the line x = e, and the x-axis. When we rotate this region about the x-axis, it forms a solid with a hole in the center.
To set up the integral, we need to express the volume of a single cylindrical shell. The volume of a cylindrical shell can be calculated by multiplying its circumference by its height and thickness. Let's denote the thickness of the shell as Δx. Therefore, the volume of the shell is approximately:
- 2πr * h * Δx, where r is the radius of the shell, and h is the height of the shell.
To express r in terms of x, we need to imagine a vertical line that passes through the shell. This line will have a distance of x from the y-axis, which is the same as the radius of the shell. Since we are rotating around the x-axis, the height of the shell is f(x) - 0, because the distance from the x-axis to the function f(x) is its height. Therefore, r = x, and h = f(x) - 0 = ln x.
Now, we can express the volume of a single shell as:
- 2πx * ln x * Δx.
To find the total volume of the solid, we need to add up all the volumes of the cylindrical shells as Δx approaches zero. This can be done using an integral. Therefore, the integral representing the volume of the solid is:
- ∫ (2πx * ln x) dx
Finally, we can evaluate this integral to find the actual volume of the solid.
In summary, the integral representing the volume of the solid when region R is rotated about the x-axis is:
- ∫ (2πx * ln x) dx
Note: Don't forget to evaluate this integral to find the actual volume of the solid.