Conditions before computation
Locate intersections before choosing bounds. On each subinterval, test which curve lies above or to the right. A geometric area is nonnegative, but that does not mean taking the absolute value only after one unsplit integral always works. The integral average divides signed accumulated output by interval length. In an inverse average-value problem, the unknown endpoint occurs both in the integral bounds and in that length.
Worked example with a method check
Find the total area between y=x² and y=9 for 0≤x≤6. The curves cross at x=3 in the interval; the horizontal line is above before that point. Integrate (9−x²) on [0,3] and (x²−9) on [3,6]. The pieces are 18 and 36; the total is 54.
Compare two methods
For y=x² and y=1 on [0,2], the signed integral of x²−1 is 2/3, but the total area is 2 because the negative portion on [0,1] must contribute positively. For x=y² and x=2y, horizontal slices give a simple width 2y−y² on [0,2]; solving each curve for y is unnecessary and can complicate the description.
Transfer: solve before reading
Find the area between x=y² and x=3y. Then find the positive endpoint b for which the average of 3x² on [0,b] is 16. Explain why b is not 16.
Transfer reasoning and answer
Intersections are y=0 and 3; integrating 3y−y² gives 27/2−9=9/2. The average of 3x² on [0,b] is b², so b=4. The number 16 is an output average, while b is an input endpoint with a different role.
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.
Find the total area between y=x² and y=9 for 0≤x≤6.
- The curves cross at x=3 in the interval; the horizontal line is above before that point.
- Integrate (9−x²) on [0,3] and (x²−9) on [3,6].
- The pieces are 18 and 36; the total is 54.
Compare the other answer choices
Use the signed net integral instead of total area.
Include only the second subregion.
Integrate x² alone and ignore the horizontal boundary.
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