AP Calc BC Integration and Accumulation of Change

Question 1

What is the value of ∫ from 0 to 1 of x·eˣ dx?

Accepted Answer

1

Answer

Integration by parts with u = x and dv = eˣ dx gives x·eˣ − eˣ. Evaluating from 0 to 1 gives (e − e) − (0 − 1) = 1.

Question 2

Which expression equals ∫ 1/((x − 1)(x + 2)) dx?

Accepted Answer

(1/3)·ln|(x − 1)/(x + 2)| + C

Answer

Partial fractions give 1/((x−1)(x+2)) = (1/3)/(x−1) − (1/3)/(x+2). Integrating gives (1/3)ln|x−1| − (1/3)ln|x+2| + C.

Question 3

If F(x) = ∫ from 0 to x² of √(1 + t³) dt, what is F′(x)?

Accepted Answer

2x·√(1 + x⁶)

Answer

By the Fundamental Theorem of Calculus with the chain rule, F′(x) = √(1 + (x²)³) · d/dx(x²) = 2x·√(1 + x⁶).

Question 4

A left Riemann sum with 4 subintervals of equal width is used to approximate ∫ from 0 to 4 of x² dx. What is its value?

Accepted Answer

14

Answer

The width is 1 and the left endpoints are 0, 1, 2, 3. The sum is 1·(0 + 1 + 4 + 9) = 14.

Question 5

What is the value of ∫ from 0 to ∞ of e^(−2x) dx?

Accepted Answer

1/2

Answer

The integral is the limit as b→∞ of [−(1/2)e^(−2x)] from 0 to b. That is 0 − (−1/2) = 1/2.

AP Calc BC Practice Test