Volume integrals let you find the size of a three-dimensional solid by adding many thin slices. The main skill is not just integrating, but setting up the correct slice, thickness, and limits. A good diagram turns the problem into a clear plan by showing the bounded region and one representative cross section.
This matters in geometry, physics, engineering, and any situation where a changing cross section builds a solid.
To set up the integral, first decide whether slices are vertical or horizontal. Vertical slices use dx and usually run from x = a to x = b, while horizontal slices use dy and usually run from y = c to y = d. The slice length, radius, or cross-sectional area must be written in terms of the same variable as the thickness.
Once the area of one slice is found, the volume is V = integral of A times thickness over the correct interval.
Key Facts
- General slicing formula: V = integral from a to b of A(x) dx
- Using horizontal slices: V = integral from c to d of A(y) dy
- Disk method about the x-axis: V = pi integral from a to b of [R(x)]^2 dx
- Washer method about the x-axis: V = pi integral from a to b of ([R(x)]^2 - [r(x)]^2) dx
- For vertical slices, thickness is dx and all slice measurements must be functions of x
- For horizontal slices, thickness is dy and all slice measurements must be functions of y
Vocabulary
- Representative slice
- A representative slice is one thin piece of the region used to model the cross section of the solid.
- Thickness
- Thickness is the small width of a slice, written as dx for vertical slices or dy for horizontal slices.
- Cross-sectional area
- Cross-sectional area is the area A(x) or A(y) of one slice of the solid before it is added into the integral.
- Washer
- A washer is a circular cross section with a hole, so its area is pi times outer radius squared minus inner radius squared.
- Limits of integration
- Limits of integration are the starting and ending values of the variable that cover the entire bounded region.
Common Mistakes to Avoid
- Choosing dx but writing functions of y is wrong because the slice thickness and slice formula must use the same variable.
- Using the top curve minus bottom curve as a radius without checking the axis of rotation is wrong because radius is a distance from the axis, not just a vertical gap.
- Forgetting the inner radius in a washer problem is wrong because a hole must be subtracted from the outer disk area.
- Using intersection points from the wrong variable is wrong because vertical slices need x-limits and horizontal slices need y-limits.
Practice Questions
- 1 The region between y = x and y = x^2 from x = 0 to x = 1 is revolved about the x-axis. Set up the volume integral using washers.
- 2 The region under y = 4 - x^2 and above the x-axis from x = -2 to x = 2 is revolved about the x-axis. Write the disk-method integral for the volume.
- 3 A region is easier to describe as x = y^2 on the left and x = 4 on the right for 0 <= y <= 2. Explain why horizontal slices with dy are a better choice than vertical slices with dx.