Applications of Integration Test – AP Calculus AB Math (Printable PDF)

Question 1 Average value (2 points)

What is the average value of f(x) = 3x² on [1, 3]?

1. 13
2. 26
3. 8
4. 9

Question 2 Area between curves (2 points)

Which value equals the area trapped between y = 2x and y = x²?

1. 4/3
2. 8/3
3. 2
4. 4

Question 3 Disc method (3 points)

The region bounded by y = x³, the x-axis, x = 1 and x = 2 is rotated around the x-axis.

1. Write an integral for the volume. (2 points)
2. Evaluate it. (1 point)

Question 4 Washer method (3 points)

Let R be the region enclosed by y = 5x and y = x².

1. Find the intersection points. (1 point)
2. Write an integral for the volume when R is rotated around the x-axis. (1 point)
3. Evaluate it. (1 point)

Question 5 Cross sections (3 points)

A museum model rests on the base under y = √x for 0 ≤ x ≤ 4. Vertical slices taken at right angles to the x-axis are equilateral triangles standing on the base.

1. Find the area of a cross section at x. (1 point)
2. Write an integral for the volume. (1 point)
3. Find the volume. (1 point)

Question 6 Displacement and distance (3 points)

A cart rolls along a track with velocity v(t) = t² − 6t + 5 over 0 ≤ t ≤ 6.

1. Find the net change in position. (1 point)
2. Find the total path length. (2 points)

Question 7 Rates in and out (2 points)

A reservoir begins with 50 gallons. During the next 8 minutes, a hose adds I(t) = 3t gallons each minute while a pump removes a steady 9 gallons each minute.

1. How many gallons are in the tank at t = 8? (1 point)
2. Is the amount increasing or decreasing at t = 2? Justify. (1 point)

Question 8 Area with respect to y (2 points)

Two boundaries, x = y² and x = y + 2, enclose the shaded region shown. Set up and evaluate a y-integral for its area.