Integral calculus: accumulation and method selection assessment
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 140 minutes, as an unvalidated planning estimate. This collection contains 10 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose methods without opening the worked solutions first. Use paper for diagrams and calculations; write enough reasoning to justify assumptions and every requested conclusion.
The suggested working time is guidance, not a timer. Written and symbolic responses require human review; a recorded response is not a proof-quality grade.
This public mixed practice samples the published unit. It is not a secure examination, complete AP paper or guarantee of course readiness. Saved first responses and known answer exposure persist across retries and imports.
Loading saved assessment responses…
0 of 10 tasks have recorded responses. No overall mastery score is assigned.
For f(x)=3−x on [0,2], compute left and right sums with four equal subintervals and the exact integral. Explain which bounds the integral above and below, and why the comparison requires more than an arbitrary table of samples.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Define F(x)=∫ from x to x² of (1+t²) dt for all real x. Derive F′(x) using FTC and also by evaluating the integral. Explain the orientation when0<x<1.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Evaluate ∫ from0 to2 of x√(x²+5) dx exactly. Show a substitution including new bounds, then verify the antiderivative by differentiation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A particle has velocity v(t)=t²−4 on0≤t≤3 seconds, in m/s, and initial position5 m. Find its final position and total distance traveled. Interpret the corresponding signed and unsigned areas under the velocity graph.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Rotate the region bounded by y=x² and y=4 about the y-axis. Set up and evaluate the volume by horizontal disks and independently by vertical shells. Explain why integrating shells over both negative and positive x would double-count.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Find the average value of f(x)=x ln x on[1,e]. Derive the needed primitive by parts and state why the average is attained somewhere in the interval.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Decompose1/(x²−1) into partial fractions and give a primitive on intervals excluding±1. Decide whether its ordinary improper integral over[0,2] converges; explain why direct endpoint subtraction is invalid.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Evaluate ∫₀^(π/2) sin x cos²x dx. Separately, transform ∫₀² dx/√(16−x²) by x=4sinθ, giving the branch, new bounds and result. Explain why √(16cos²θ) cannot be replaced by4cosθ on every real θ.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Without finding an elementary primitive, prove that ∫₁^∞1/(x²+3) dx converges and bound it between1/4 and1. State why you may pass from finite-interval comparisons to this conclusion.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For composite trapezoids on[0,3], assume f has a continuous second derivative with |f″|≤8. Using the standard error bound, what is the smallest positive integer n of equal subintervals that guarantees absolute error at most1/8?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
For f(x)=3−x on [0,2], compute left and right sums with four equal subintervals and the exact integral. Explain which bounds the integral above and below, and why the comparison requires more than an arbitrary table of samples.
A particle has velocity v(t)=t²−4 on0≤t≤3 seconds, in m/s, and initial position5 m. Find its final position and total distance traveled. Interpret the corresponding signed and unsigned areas under the velocity graph.
Rotate the region bounded by y=x² and y=4 about the y-axis. Set up and evaluate the volume by horizontal disks and independently by vertical shells. Explain why integrating shells over both negative and positive x would double-count.
Decompose1/(x²−1) into partial fractions and give a primitive on intervals excluding±1. Decide whether its ordinary improper integral over[0,2] converges; explain why direct endpoint subtraction is invalid.
Evaluate ∫₀^(π/2) sin x cos²x dx. Separately, transform ∫₀² dx/√(16−x²) by x=4sinθ, giving the branch, new bounds and result. Explain why √(16cos²θ) cannot be replaced by4cosθ on every real θ.
Without finding an elementary primitive, prove that ∫₁^∞1/(x²+3) dx converges and bound it between1/4 and1. State why you may pass from finite-interval comparisons to this conclusion.
For composite trapezoids on[0,3], assume f has a continuous second derivative with |f″|≤8. Using the standard error bound, what is the smallest positive integer n of equal subintervals that guarantees absolute error at most1/8?
