Conditions before computation
Left sums use the left value of each adjacent interval; the final table value supplies no extra left rectangle. Trapezoids average the two endpoint heights for each interval separately. The function’s monotonicity can bound left and right estimates, while convexity can establish a trapezoid overestimate. A table alone does not establish behavior between samples unless the problem supplies it. Keep an estimate distinct from an exact integral and from the error between them.
Worked example with a method check
On the same interval [p,q], ∫f(x)dx=3 and ∫g(x)dx=-6. Find ∫[3f(x)−2g(x)]dx on [p,q]. Integration is linear for integrable functions over the same interval. Apply the coefficients to the signed integral values: 3(3)−2(-6). The result is 21.
Compare two methods
For samples at 0,1,3, a sum with one common width is unjustified. If rates are 2,5,8, the left estimate is 1·2+2·5=12, while the trapezoid estimate is 1·(2+5)/2+2·(5+8)/2=33/2. Neither table estimate is guaranteed exact for an arbitrary intervening curve. If the curve is explicitly piecewise linear between samples, the trapezoids do give the exact signed integral.
Transfer: solve before reading
Suppose ∫₀²f=7 and ∫₀⁵f=4. Find ∫₅²f. Then compute the trapezoid error for x² on [0,3] using nodes 0,1,3, and justify its direction from shape.
Transfer reasoning and answer
The forward missing integral is ∫₂⁵f=4−7=−3, so reversing gives ∫₅²f=3. The trapezoid estimate is 21/2 and the exact integral is 9, leaving an excess of 3/2. Since x² is convex, its secant segments lie above the curve, agreeing with the positive excess.
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.
On the same interval [p,q], ∫f(x)dx=3 and ∫g(x)dx=-6. Find ∫[3f(x)−2g(x)]dx on [p,q].
- Integration is linear for integrable functions over the same interval.
- Apply the coefficients to the signed integral values: 3(3)−2(-6).
- The result is 21.
Compare the other answer choices
Drop both coefficients.
Omit the coefficient 2 on g.
Change subtraction of the signed second integral into addition.
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