AP Calc AB Applications of Integration

Question 1

What is the average value of f(x) = sin x on the interval [0, π]?

Accepted Answer

2/π

Answer

The average value is (1/(π−0))∫₀^π sin x dx = (1/π)(2) = 2/π.

Question 2

A particle moves along a line with velocity v(t) = t² − 4t + 3 for 0 ≤ t ≤ 3. What is the total distance the particle travels on this interval?

Accepted Answer

8/3

Answer

v(t) = (t−1)(t−3) is positive on (0,1) and negative on (1,3). The distance is ∫₀¹ v dt − ∫₁³ v dt = 4/3 + 4/3 = 8/3.

Question 3

What is the area of the region enclosed by the graphs of y = x and y = x²?

Accepted Answer

1/6

Answer

The graphs intersect at x = 0 and x = 1, and y = x lies above y = x² between them. The area is ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6.

Question 4

The region bounded by y = √x, the x-axis, and the line x = 4 is revolved about the x-axis. What is the volume of the resulting solid?

Accepted Answer

8π

Answer

By the disk method, V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π(16/2) = 8π.

Question 5

The region enclosed by y = x and y = x² is revolved about the x-axis. What is the volume of the solid generated?

Accepted Answer

2π/15

Answer

Use washers with outer radius x and inner radius x²: V = π∫₀¹ (x² − x⁴) dx = π(1/3 − 1/5) = 2π/15.

AP Calc AB Practice Test