Volumes with Known Cross Sections
- Set up a volume integral for solids with square, rectangular, triangular, or semicircular cross sections
- Identify the base region and express the cross-sectional side in terms of it
- Distinguish cross-section volumes from disk and washer revolutions
One idea, applied to any shape
Every volume in this unit comes from the same integral: V = ∫ A(x)dx, where A(x) is the area of the cross section at position x. The whole problem is finding A(x), and that reduces to finding s, the length of the cross section's side, from the base region. If cross sections are perpendicular to the x-axis and the base lies between curves y = f(x) and y = g(x), then s = f(x) − g(x) — the same top-minus-bottom expression used for area. Then substitute into the area formula for the named shape. Notice that disks and washers are not a separate topic: a disk is the case A = π r², and knowing the general form means never having to remember which formula goes with which situation.
The four shapes worth having memorized
With s the cross-sectional side taken from the base region: a square gives A = s². A rectangle of height twice the base gives A = 2s², and any stated ratio scales the same way. An equilateral triangle gives A = (√3/4)s². A semicircle with s as its diameter gives A = (π/8)s², because the radius is s/2 and A = (1/2)π(s/2)² = πs²/8. That last one is where most errors happen: the problem gives you a diameter and the formula wants a radius, so the factor is πs²/8 rather than πs²/2. Read carefully whether s is described as the diameter or the radius — the exam varies it deliberately.
Which axis, and which variable
The phrase "cross sections perpendicular to the x-axis" means the slices are vertical planes, so s is measured vertically as a function of x and you integrate dx over an interval of x-values. "Perpendicular to the y-axis" means horizontal slices, so s is measured horizontally — right curve minus left curve, expressed as functions of y — and you integrate dy over an interval of y-values. This is the same decision as the dx-or-dy choice for area, with the same consequence: getting it wrong produces an integral in one variable with limits from the other, which is not a computation that can be salvaged. Write down which axis before setting anything up.
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.
- 1.Identify the base region: it runs from x = 0 to x = 4, bounded above by y = √x and below by y = 0.
- 2.The cross-sectional side at position x is the vertical extent of the base: s = √x − 0 = √x.
- 3.For a square, A(x) = s² = (√x)² = x.
- 4.Set up the volume: V = ∫₀⁴ x dx.
- 5.Antidifferentiate: x²/2. Evaluate: 4²/2 − 0 = 16/2 = 8.
For a semicircular cross section, check whether s is the diameter or the radius before writing anything. Diameter s gives A = πs²/8; radius s gives A = πs²/2. The factor of four between them is the most frequently missed detail in this topic.
A solid has base bounded by y = f(x) and y = 0 on [0, 3], with cross sections perpendicular to the x-axis that are equilateral triangles. The volume is
For cross sections perpendicular to the y-axis, the side s should be expressed as
Answer the 2 checkpoints as you read.
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