AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/8/8-7)
Unit 8 · Topic 8.7
8.7 Volumes with Cross Sections: Squares and Rectangles
Some solids have a flat base region, and every slice perpendicular to an axis has the same shape. If you can find the area of one slice, A(x), the volume is the integral of A(x). This topic covers square and rectangle cross sections.
Key terms
- cross section
- perpendicular
- base region
- area of a slice
- volume
The slicing idea
Picture a loaf of bread. Each slice is thin, with thickness dx, and has some cross-sectional area A(x). Its volume is about A(x)·dx. Add up all the slices from one end of the loaf to the other:
Volume = ∫ₐᵇ A(x) dx.
In cross-section problems, the base of the solid is a flat region in the xy-plane, and slices perpendicular to the x-axis (or y-axis) are squares, rectangles, triangles or semicircles that stand up out of the plane.
Finding the side length
The slice's side that lies in the base is the length of the base region at that x. That's just the top − bottom distance from area problems:
s(x) = top(x) − bottom(x) for slices perpendicular to the x-axis,
s(y) = right(y) − left(y) for slices perpendicular to the y-axis.
Then plug s into the shape's area formula. A square with side s has area s². A rectangle whose height is h times its base has area s·(h·s) = h·s². If the height is a fixed number H, the area is s·H.
The steps
- Sketch the base region and one slice.
- Decide the direction of slicing: perpendicular to the x-axis means integrate with respect to x; perpendicular to the y-axis means integrate with respect to y.
- Write s as top − bottom (or right − left).
- Write the area A in terms of s, then in terms of x (or y).
- Integrate A over the base's extent: ∫ₐᵇ A(x) dx.
Cross sections from a table
Sometimes you're given the area of the cross sections at a few points instead of a formula. For example, a table might list A(x), the area in square feet of a log's cross section x feet from one end. Then the volume ∫ A(x) dx can be estimated with a Riemann or trapezoidal sum (6.2), exactly like any other integral from a table. The units are square feet × feet = cubic feet.
What doesn't change
There's no π here unless the slices are circular. And notice that the volume doesn't depend on whether the squares stand up or hang down: only the area of each slice matters. If a question says the slices are perpendicular to the y-axis, every piece of the setup is in y, including the limits.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Square cross sections
The base of a solid is the region bounded by y = √x, y = 0 and x = 4. Cross sections perpendicular to the x-axis are squares. Find the volume.
Show the solutionHide the solution
- Step 1: Side length at x: s = √x − 0 = √x.
- Step 2: Area of a square slice: A(x) = (√x)² = x.
- Step 3: The base runs from x = 0 to x = 4.
- Step 4: Volume = ∫₀⁴ x dx = 16/2 = 8.
Answer: 8 cubic units
- Example 2
Rectangles with height tied to the base
The base of a solid is the region between y = 4 − x² and the x-axis. Cross sections perpendicular to the x-axis are rectangles whose height is half the base. Find the volume.
Show the solutionHide the solution
- Step 1: Base of each rectangle: s = 4 − x², from x = −2 to x = 2.
- Step 2: Height = s/2, so area = s · s/2 = (4 − x²)²/2.
- Step 3: Volume = ½ ∫ from −2 to 2 of (4 − x²)² dx = ½ ∫ from −2 to 2 of (16 − 8x² + x⁴) dx.
- Step 4: ∫ from −2 to 2 of (16 − 8x² + x⁴) dx = 2[16(2) − 8(8)/3 + 32/5] = 2(32 − 64/3 + 32/5) = 512/15.
- Step 5: Volume = 256/15 ≈ 17.067.
Answer: 256/15 cubic units
- Example 3
Trap: slices perpendicular to the y-axis
The base of a solid is the region bounded by y = x² and y = 4. Cross sections perpendicular to the y-axis are squares. Find the volume.
Show the solutionHide the solution
- Step 1: Slices perpendicular to the y-axis are horizontal strips, so work in y.
- Step 2: At height y, the region runs from x = −√y to x = √y, so s = 2√y. (Using s = √y misses half of the region.)
- Step 3: Area = (2√y)² = 4y.
- Step 4: y runs from 0 to 4. Volume = ∫₀⁴ 4y dy = 2y² from 0 to 4 = 32.
Answer: 32 cubic units
Common mistakes
- Squaring each function separately: (f − g)² is not f² − g².
- Putting a π in a square cross-section volume.
- Using the wrong variable for the slicing direction.
- Using only the right half of a region that's symmetric about the y-axis.
On the exam
- Free-response region questions often end with a cross-section volume. Getting the setup ∫ (top − bottom)² dx right is the key step.
- Read whether the height is “equal to,” “twice” or “half” the base. That number multiplies s².
Connected topics
Videos
Check yourself
4 questions on 8.7 Volumes with Cross Sections: Squares and Rectangles. Pick an answer to see if you got it, and why.
The base of a solid is the region bounded by the graphs of y = x and y = x². Each cross section of the solid perpendicular to the x-axis is a rectangle whose height is 3 times the length of its base. What is the volume of the solid?
The base of a solid is the region bounded by the graph of y = √x, the x-axis, and the line x = 4. Each cross section of the solid perpendicular to the x-axis is a square. What is the volume of the solid?
The base of a solid is the region bounded by the graph of y = x² and the line y = 4. Each cross section of the solid perpendicular to the y-axis is a square. What is the volume of the solid?
The base of a solid is the region bounded by the graphs of y = 2 − x² and y = x². Each cross section of the solid perpendicular to the x-axis is a rectangle whose height is half the length of its base. What is the volume of the solid?
0 of 4 answered