AP Calculus BC Unit 6 review: Integration and Accumulation of Change

• Dropping the chain rule. College Board's course description warns about it twice: when differentiating a function defined by an integral, and when doing u-substitution.
• Keeping x-limits after switching to u. On a definite integral, change the limits when you substitute, or switch back to x before plugging in.
• Leaving off + C. The course description expects a constant of integration on every indefinite integral.
• Giving a calculator answer with no setup. On free response, write the definite integral with its limits and the dx before the number your calculator gives.
• Chains of equal signs. Writing expressions that aren't equal on one line joined by "=", or losing parentheses around a sum, is flagged in the course description.
• Using one width for an uneven table. If the times are 0, 2, 5 and 9, the intervals are 2, 3 and 4 wide.
• Forgetting the starting amount. An integral of a rate gives the change. The amount is the starting value plus that change.
• Missing an improper integral. ∫ from −1 to 1 of 1/x² looks like a normal integral, but the integrand is unbounded at 0 and the integral diverges. Plugging in the limits gives −2, a negative answer for a positive function.

1. Question 1 · 6.1 and 6.7

   A tank holds 20 liters at t = 0. Water leaks out at r(t) = 6 − 2t liters per minute for 0 ≤ t ≤ 3. How much water is in the tank at t = 3?

   • A9 liters
   • B14 liters
   • C29 liters
   • D11 liters
   Show the answer

   D. Amount left = starting amount − amount leaked = 20 − ∫₀³ (6 − 2t) dt = 20 − [6t − t²]₀³ = 20 − 9 = 11 liters. 9 liters is the amount that leaked. 29 adds the leak instead of subtracting it. Liters per minute times minutes gives liters, which is how you know the integral is an amount.

2. Question 2 · 6.2 Riemann sums

   A runner's velocity v(t), in meters per second, is measured at four times. t (seconds): 0, 2, 5, 9 v(t) (m/s): 4, 7, 6, 3 Using a trapezoidal sum with the three intervals in the table, estimate the distance the runner covers from t = 0 to t = 9.

   • A44 meters
   • B48.5 meters
   • C49.5 meters
   • D53 meters
   Show the answer

   B. Use each interval's own width: ½(4 + 7)(2) + ½(7 + 6)(3) + ½(6 + 3)(4) = 11 + 19.5 + 18 = 48.5 meters. 49.5 comes from treating every interval as 3 seconds wide. 53 and 44 are the left and right Riemann sums.

3. Question 3 · 6.4 FTC and accumulation functions

   Let g(x) = ∫₂^(x³) √(1 + t²) dt. What is g′(1)?

   • A3√2
   • B√2
   • C3√2 − √5
   • D3
   Show the answer

   A. By the Fundamental Theorem with the chain rule, g′(x) = √(1 + (x³)²) · 3x². At x = 1 that's √2 · 3 = 3√2. √2 forgets the chain-rule factor 3x². The lower limit is a constant, so it adds nothing to the derivative, which rules out 3√2 − √5.

4. Question 4 · 6.9 Substitution

   Evaluate ∫₀² x e^(x²) dx.

   • Ae⁴ − 1
   • B½(e² − 1)
   • C½(e⁴ − 1)
   • D2e⁴
   Show the answer

   C. Let u = x², so du = 2x dx and x dx = ½ du. The limits change too: x = 0 gives u = 0, and x = 2 gives u = 4. The integral becomes ½ ∫₀⁴ eᵘ du = ½(e⁴ − 1). ½(e² − 1) keeps the old upper limit after switching to u. e⁴ − 1 drops the ½.

5. Question 5 · 6.11 Integration by parts (BC)

   Find ∫ x cos x dx.

   • Ax sin x + cos x + C
   • Bx sin x − cos x + C
   • C½x² sin x + C
   • D−x sin x + cos x + C
   Show the answer

   A. Take u = x and dv = cos x dx, so du = dx and v = sin x. Then ∫ x cos x dx = x sin x − ∫ sin x dx = x sin x + cos x + C. Check by differentiating: sin x + x cos x − sin x = x cos x. The answer with − cos x gets the sign of ∫ sin x dx wrong.

6. Question 6 · 6.12 Partial fractions (BC)

   Find ∫ 5 / ((x − 2)(x + 3)) dx.

   • Aln|x − 2| + ln|x + 3| + C
   • Bln|x − 2| − ln|x + 3| + C
   • C5 ln|(x − 2)(x + 3)| + C
   • Dln|x + 3| − ln|x − 2| + C
   Show the answer

   B. Write 5 / ((x − 2)(x + 3)) = A / (x − 2) + B / (x + 3), so 5 = A(x + 3) + B(x − 2). Setting x = 2 gives 5 = 5A, so A = 1. Setting x = −3 gives 5 = −5B, so B = −1. Each piece integrates to a natural log: ln|x − 2| − ln|x + 3| + C.

7. Question 7 · 6.13 Improper integrals (BC)

   Which of these improper integrals converges?

   • A∫₁^∞ 1/x dx
   • B∫₁^∞ 1/√x dx
   • C∫₁^∞ 1/x² dx
   • D∫₀¹ 1/x dx
   Show the answer

   C. ∫₁ᵇ 1/x² dx = 1 − 1/b, which approaches 1 as b → ∞, so it converges. ∫₁ᵇ 1/x dx = ln b and ∫₁ᵇ 1/√x dx = 2√b − 2 both grow without bound. ∫₀¹ 1/x dx is improper at 0 instead, and it diverges too: ∫ₐ¹ 1/x dx = −ln a grows without bound as a → 0⁺.

8. Question 8 · 6.6 Properties of definite integrals

   ∫₁⁵ f(x) dx = 8 and ∫₃⁵ f(x) dx = 3. What is ∫₁³ (2f(x) + 1) dx?

   • A10
   • B11
   • C22
   • D12
   Show the answer

   D. Split the interval: ∫₁³ f(x) dx = 8 − 3 = 5. Then ∫₁³ (2f(x) + 1) dx = 2(5) + ∫₁³ 1 dx = 10 + 2 = 12. 11 adds 1 instead of integrating it over an interval 2 wide. 10 forgets the + 1 part altogether.

Start with accumulation, not techniques. Most free-response questions in this unit hand you a rate, from a formula, a table or a graph, and ask for an amount, so practice reading every integral with its units. Then drill u-substitution until it's automatic. The course description recommends exactly that, followed by mixed practice where you have to pick the technique yourself.

Do most of it without a calculator. Multiple-choice Part A (29 questions) and free-response Part B (4 questions) don't allow one. When you do use a calculator, practice writing the setup first, the way graders expect.

For more problems, Khan Academy's AP Calculus BC unit on integration follows the same topic numbers. OpenStax's free textbooks cover the theory: Calculus Volume 1, chapter 5 for Riemann sums and the Fundamental Theorem, and Calculus Volume 2, chapter 3 for parts, partial fractions and improper integrals. The topic list here comes from College Board's AP Calculus AB and BC course and exam description.

Ashore's Calc BC course has a lesson and practice questions for every topic in this unit. The free starting check takes about 15 minutes and shows which ones are costing you points.

• AP Calculus BC: all 10 units, exam format and scores