Short on time? This page is the recap. For the derivations, the figure and the worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: ∫ (a to b) A(x) dx means the definite integral of A(x) from x = a to x = b. s is the side of the slice that lies in the base.
Recap
- The method is Topic 8.7’s: V = ∫ (a to b) A(x) dx, built from thin slabs of volume A(x) Δx.
- Perpendicular to the x-axis: s(x) = top − bottom, x-limits, dx.
- Perpendicular to the y-axis: s(y) = right − left in y, y-limits, dy.
- Each standard shape has A = c s², so V = c ∫ s² dx.
- If the area A(x) is given as a function, integrate it directly.
Key relationships
| Slice on side s | Area | Where it comes from |
|---|---|---|
| Equilateral triangle | (√3/4)s² | height (√3/2)s by Pythagoras |
| Isosceles right triangle, leg in base | s²/2 | legs s and s |
| Isosceles right triangle, hypotenuse in base | s²/4 | legs s/√2, or height s/2 |
| Triangle, base s, height h | (1/2)sh | height given in the question |
| Semicircle, diameter in base | (π/8)s² | radius s/2 |
| Circle, diameter in base | (π/4)s² | radius s/2 |
Ratio check: equilateral ÷ semicircle = (√3/4) ÷ (π/8) = 2√3/π ≈ 1.103 for every base.
Assumptions behind the method
- Every slice stands at right angles to the base and perpendicular to the stated axis.
- The side s is a length across the base, so s ≥ 0 between the limits.
- One formula for s holds across the interval; otherwise split the integral.
Mistakes to avoid
- Semicircle radius = s. It is s/2. Using s makes the answer four times too big.
- Swapping the right-triangle cases: leg s²/2, hypotenuse s²/4.
- Writing √3/2 for the equilateral constant. The area constant is √3/4.
- Writing f² − g² instead of (f − g)².
- Mixing x and y in a dy integral.
- Rounding a calculator limit before integrating.
Quick self-check
- A semicircular slice has diameter 6. What is its area? (9π/2: radius 3, so (1/2)π(3)².)
- An equilateral triangular slice has side 2. What is its area? (√3: (√3/4)(4).)
- Two solids share a base. One has isosceles right triangles with the hypotenuse in the base, the other with a leg in the base. What is the ratio of their volumes? (1 : 2, since (1/4) ÷ (1/2) = 1/2.)
Next: practice questions.