Conditions before computation
For a washer, subtract inner squared radius from outer squared radius after measuring both distances from the actual axis of rotation. The square of a radius difference is not the difference of squared radii. In a semicircular cross section, a base segment described as a diameter must be halved before squaring. In an equilateral triangle, derive altitude as √3 times half the side and then apply the one-half base-times-height formula.
Worked example with a method check
A region between y=3 and y=6, for 0≤x≤6, rotates about the x-axis. Its volume is kπ. Find k. The outer radius is 6; the inner radius is 3. The washer area divided by π is (6)²−3²=27. Multiply by length 6: k=162.
Compare two methods
Rotating the region 0≤y≤x about y=−2 gives outer radius x+2 and inner radius 2, so the area is π[(x+2)²−4]. Rotating about y=0 instead gives πx². The unchanged base region does not imply unchanged volume when the axis moves. For nonrotational triangular sections, inserting π would describe the wrong cross-sectional shape entirely.
Transfer: solve before reading
For 0≤x≤2, semicircular sections have diameter 2√x. Find the volume, and compare it with the volume obtained by mistakenly treating 2√x as the radius.
Transfer reasoning and answer
The correct radius is √x, so the area is πx/2 and its integral is π. Treating the diameter as radius makes each area four times as large and yields 4π. Halving a length before squaring changes area by a factor of four, not two.
Practice the next representation
Work all four task families linked to this lesson. For the illustrated items, state which numerical labels and relationships you used. For a written extension, compare each line of your solution with its original self-assessment criteria; these criteria are practice feedback, not an official AP score.
A region between y=3 and y=6, for 0≤x≤6, rotates about the x-axis. Its volume is kπ. Find k.
- The outer radius is 6; the inner radius is 3.
- The washer area divided by π is (6)²−3²=27.
- Multiply by length 6: k=162.
Compare the other answer choices
Use disks and fail to subtract the hole.
Add the inner and outer squared radii.
Subtract the outer squared radius from the inner one.
Explain your thinking.
Optional written extension. Self-assessed only; it does not change your automatic practice result. These are teaching criteria, not official awarded marks.
(a) State the relationship or theorem that makes this calculation valid, including its relevant conditions. (b) Show the intermediate computation and give the requested result. (c) Explain why one plausible competing method would fail for this problem.
This draft stays on this page only. Use paper if you want to keep your working.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB and BC Course and Exam DescriptionCalculus Volume 1: Fundamental Theorem of Calculus