AP Calc AB Integration and Accumulation of Change

Question 1

What is the value of ∫ from 0 to 2 of (3x² − 2x) dx?

Accepted Answer

4

Answer

An antiderivative is x³ − x². Evaluating from 0 to 2 gives (8 − 4) − 0 = 4.

Question 2

If G(x) = ∫ from 1 to x² of sin(t) dt, what is G′(x)?

Accepted Answer

2x·sin(x²)

Answer

By the Fundamental Theorem of Calculus with the chain rule, G′(x) = sin(x²)·d/dx(x²) = 2x·sin(x²).

Question 3

Which of the following is equal to ∫ 2x·cos(x²) dx?

Accepted Answer

sin(x²) + C

Answer

Let u = x², so du = 2x dx. The integral becomes ∫ cos u du = sin u + C = sin(x²) + C.

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 the value of the sum?

Accepted Answer

14

Answer

Each 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 1 of x·e^(x²) dx?

Accepted Answer

(e − 1)/2

Answer

Let u = x², so du = 2x dx and x dx = du/2. The integral becomes (1/2)∫ from 0 to 1 of e^u du = (e − 1)/2.

AP Calc AB Practice Test