A reliable method
For area between curves use upper minus lower, splitting at intersections. Divide an integral by interval length for average value. Disk volumes use π times radius squared.
Worked foundation example
Find the area between y=2x and y=0 from x=0 to x=3. The upper function is the line throughout this nonnegative interval. Integrate 2x from 0 to 3. The area is 9 square units.
The common trap
The square of an integral is not the integral of a square. Average value divides by interval length, not by the number of visible terms.
Connect the representations
Check units: integrating a speed over time gives distance, and integrating a cross-sectional area over length gives volume.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare area, average value and volume
For y=2x on [0,3], area under the line is 9. The average function value is area divided by interval width, 9/3=3. Rotating the same region about the x-axis requires disk area π(2x)², giving volume 36π. The same curve produces three different outputs because each question accumulates a different quantity. Squaring the radius is essential for volume.
Transfer and explain
Transfer: the region between y=x+1 and y=x on [0,2] has area 2, since upper minus lower is 1. A disk formula using only the upper curve would not compute this planar area; identify the requested geometry first.
Find the area between the line segment shown and the x-axis from x=0 to x=3.
Open full-size diagram ↗- The upper function is the line throughout this nonnegative interval.
- Integrate 2x from 0 to 3.
- The area is 9 square units.
Compare the other answer choices
Omit the interval width in the triangle area.
Report the endpoint height.
Use the bounding rectangle instead of triangle.
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) Find the planar area. (b) Find the average height. (c) Rotate this region about the x-axis and find the volume.
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 courseCalculus Volume 1: Fundamental Theorem of Calculus