A solid of revolution is formed when a plane region rotates around a line. A correct visualization begins with the original two-dimensional boundaries, the interval, and the axis of rotation.
Worked example
Rotate the region bounded by y=sqrt(x), y=0, x=1, and x=4 around the x-axis.
1. Create the boundary functions
In the Graphing Calculator, graph y=sqrt(x) and y=0. Restrict attention to the interval [1,4].
2. Open the solid tool
Go to the Solid of Revolution visualizer. Choose the square-root graph as the first boundary, the x-axis as the second boundary, set a=1 and b=4, and choose rotation around the x-axis.
3. Interpret the cross-sections
At each x-value, the radius is R(x)=sqrt(x). Since the inner boundary is the x-axis, the inner radius is zero. The cross-sections are disks.
The corresponding volume setup is:
V = pi * integral from 1 to 4 of (sqrt(x))^2 dx
Since (sqrt(x))^2=x, the exact volume is 15pi/2.
Washer regions
If the region lies between two nonzero curves and rotates around the x-axis, the outer radius is the farther curve and the inner radius is the nearer curve. The cross-sectional area is pi(R^2-r^2).
Rotation around the y-axis
Choose the y-axis when the intended solid rotates horizontally around x=0. Depending on how the region is described, it may be easier to use functions of y or a shell-method setup for manual calculation. The visualizer helps confirm geometry, but the integration method should match the problem.
Use the display controls
Show or hide the middle cross-section, solid fill, original region, base diameters, construction lines, and axis labels. Use separate colors for the original region and the rotated object.
Check the geometry before calculating
- Confirm that
a<b. - Ensure the selected functions bound a nonzero region.
- Check which boundary is farther from the axis.
- Make sure the axis of rotation is correct.
- Inspect whether the region crosses the axis, which can change the radius setup.
Visualization is not the proof
The image helps you understand radii and cross-sections. The exact volume still depends on a correct integral and valid assumptions. Record exact bounds and formulas before approximating.
Build the square-root solid of revolution online →
For the original plane region, review how to shade between two curves.