Volume by cross-sections is a calculus method for finding the volume of a solid when you know the shape of each thin slice. Instead of using one standard formula like the volume of a cylinder, you build the solid from many small pieces. Each slice has a cross-sectional area, and adding all the slices gives the total volume.
This method matters because it connects geometry, functions, and integration in a very visual way.
The main idea is to choose an axis, usually the x-axis, and write the area of a slice as A(x). A thin slice has approximate volume A(x) dx, so the total volume is V = integral from a to b of A(x) dx. The hardest part is often finding the side length, radius, height, or base of the cross-section from the base region.
Once A(x) is written correctly, the rest is a definite integral.
Key Facts
- General formula: V = integral from a to b of A(x) dx
- A(x) is the area of a cross-section perpendicular to the x-axis at position x.
- For square cross-sections, A(x) = s(x)^2, where s(x) is the side length.
- For semicircle cross-sections, A(x) = (1/2)pi r(x)^2, often with r(x) = d(x)/2.
- For equilateral triangle cross-sections, A(x) = (sqrt(3)/4)s(x)^2.
- If slices are perpendicular to the y-axis, use V = integral from c to d of A(y) dy.
Vocabulary
- Cross-section
- A cross-section is the flat shape made when a solid is sliced by a plane.
- Base region
- The base region is the two-dimensional area in the coordinate plane that determines the widths of the slices.
- Area function
- An area function A(x) gives the area of a slice at each x-value in the interval.
- Definite integral
- A definite integral adds infinitely many tiny quantities over an interval to produce a total amount.
- Slice thickness
- Slice thickness is the small width dx or dy used to approximate the volume of one thin slice.
Common Mistakes to Avoid
- Using the base width directly as volume, which is wrong because the width must first be converted into a cross-sectional area.
- Forgetting to square the side length for square cross-sections, which gives units of length instead of units of volume.
- Using radius when the diagram gives diameter for semicircles, which makes the area four times too large if not corrected.
- Integrating with respect to x when the slices are perpendicular to the y-axis, which uses the wrong variable and usually the wrong bounds.
Practice Questions
- 1 The base region is between y = x and y = 0 from x = 0 to x = 4. Cross-sections perpendicular to the x-axis are squares. Find the volume.
- 2 The base region is bounded by y = sqrt(x), y = 0, x = 0, and x = 9. Cross-sections perpendicular to the x-axis are semicircles with diameter equal to the vertical distance in the base. Find the volume.
- 3 A solid has a base region between y = 4 - x^2 and y = 0 from x = -2 to x = 2. Cross-sections perpendicular to the x-axis are squares. Explain how the symmetry of the base region can simplify the volume integral.