AP Calculus BC Unit 8 practice questions: Applications of Integration
8 original multiple-choice questions on AP Calc BC Unit 8, applications of integration. Answer each one in your browser and read the explanation, then review the key ideas below.
Question 1 of 8
Which integral gives the length of the curve y = f(x) from x = a to x = b?
Question 2 of 8
What is the average value of f(x) = x² on [0, 3]?
Question 3 of 8
The region under y = √x from x = 0 to x = 4 is revolved around the x-axis. What is the volume?
Question 4 of 8
What is the area between y = x and y = x² from x = 0 to x = 1?
Question 5 of 8
A particle has velocity v(t) = 2t. What is its displacement from t = 0 to t = 3?
Question 6 of 8
A particle has velocity v(t) = t − 2. What total distance does it travel from t = 0 to t = 4?
Question 7 of 8
The region between y = x and y = x² (0 ≤ x ≤ 1) is revolved around the x-axis. What is the volume?
Question 8 of 8
A tank holds 5 L at t = 0 and water flows in at r(t) = 3t L/min. How much water is in the tank at t = 2?
Key ideas in Unit 8
- The average value of f on [a, b] is (1/(b − a))∫ from a to b of f(x) dx.
- Displacement is ∫v(t) dt; total distance is ∫|v(t)| dt. Position at time t is the starting position plus the accumulated change.
- Area between curves is ∫(top − bottom) dx or ∫(right − left) dy.
- Volumes: discs and washers for solids of revolution, and known cross-sections such as squares or semicircles.
- BC adds arc length of a function: ∫ from a to b of √(1 + [f′(x)]²) dx.
More AP Calc BC practice
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Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions →
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