Calculus BC · Unit 8 · Paper 2

Applications of Integration unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1 Paper 2 Paper 3

Each paper is built from this unit’s 24 terms and is the same for everyone, so a teacher can assign “Unit 8, Paper 2” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.

Suggested time 36 min 33 points0/17 attempted

Vertical slices mean top minus bottom

Arc length of y = f(x)

Cross-section area formulas

Displacement vs total distance

Total distance from a velocity function

Average value of a function

Upper curve can switch

Arc length of a parametric curve

Choosing dx or dy

Average value divides by the width

MVT for integrals

Horizontal slices mean right minus left

Short answer 1. Define or explain: Net change from a rate in an applied context

3 pts

Short answer 2. Define or explain: Switch variables before splitting

3 pts

Short answer 3. Define or explain: Arc length comes from Pythagoras

3 pts

Short answer 4. Define or explain: Write the integral before evaluating

3 pts

Free response

9 pts

NO CALCULATOR. Answer the following questions about improper integrals and arc length.

A. Determine whether ∫₁^∞ x^(−3/2) dx converges or diverges. If it converges, find its value.

B. Determine whether ∫₀¹ x^(−1/2) dx converges or diverges. If it converges, find its value.

C. Find the length of the curve y = (2/3)x^(3/2) from x = 0 to x = 3.

D. Determine whether ∫₁^∞ (1/x) dx converges or diverges. Justify your answer, and explain how the result compares with part A.