Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.
y=ln x, y=0, x=1, x=3
Please show the steps in your work, thanks! :)
Each slab of width dx of the resulting solid is a circular disc with volume
pi (ln x)^2 dx
Integrate that function from x=1 to x=3.
You do the steps. We'll be glad to critique your work.
This may help you doing the integration:
http://www.karakas-online.de/forum/viewtopic.php?t=4026
To find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis, we will use the method of cylindrical shells.
Step 1: Draw the region and the axis of rotation.
Draw the graphs of the equations y = ln(x), y = 0, x = 1, and x = 3 on a coordinate plane. Shade the region bounded by these graphs and the x-axis. Also, draw a horizontal line to indicate the axis of rotation.
Step 2: Determine the range of integration.
The region is bounded by x = 1 and x = 3. Therefore, we will integrate with respect to x from x = 1 to x = 3.
Step 3: Define the radius and height of a typical cylindrical shell.
For each x-value within the range of integration, the radius of each cylindrical shell is given by x, as we are revolving around the x-axis. The height of each cylindrical shell is given by the difference between the two functions: y = ln(x) and y = 0.
Step 4: Set up the integral to find the volume.
The volume of each cylindrical shell is given by V = 2πrh, where r is the radius and h is the height. Hence, we can express the volume of the solid as an integral:
V = ∫[1,3] 2πx(ln(x) - 0) dx
Step 5: Evaluate the integral.
Integrate the expression with respect to x, taking into account that the natural logarithm function is the antiderivative of 1/x:
V = 2π ∫[1,3] x(ln(x)) dx
This integral can be evaluated using integration techniques such as integration by parts or substitution.
Step 6: Calculate the volume.
Evaluate the definite integral:
V = 2π ∫[1,3] x(ln(x)) dx ≈ [Calculate the definite integral]
The result of this calculation will give you the volume of the solid generated by revolving the region bounded by the given equations about the x-axis.