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Unit 3 of 5

Unit 3: Integrals

Study guide for CLEP CLEP Calculus — Unit 3: Integrals. Practice questions, key concepts, and exam tips.

58

Practice Questions

18

Flashcards

5

Key Topics

Key Concepts to Study

Riemann sums
Fundamental Theorem of Calculus
u-substitution
definite integral properties
average value of a function

Sample Practice Questions

Try these 5 questions from this unit. Sign up for full access to all 58.

Q1EASY

A tank is being filled with water at a rate of 2 cubic feet per minute. If the tank is initially empty, how many cubic feet of water will it contain after 5 minutes?

A) 5 cubic feet
B) 8 cubic feet
C) 10 cubic feet
D) 12 cubic feet
E) 6 cubic feet
Show Answer

Answer: C — The tank is being filled at a constant rate, so the amount of water in the tank increases linearly with time.

Q2MEDIUM

Find area between curves y = $x^{2}$ and y = 2x on [0,2]

A) ∫[0,2] ($x^{2}$ - 2x) dx
B) ∫[0,2] (2x - $x^{2}$) dx
C) ∫[0,2] ($x^{2}$ + 2x) dx
D) ∫[0,2] (2x + $x^{2}$) dx
E) ∫[0,2] ($x^{2}$ - 2x)^2 dx
Show Answer

Answer: B — "∫[0,2] (2x - $x^{2}$) dx" is correct because area = ∫[0,2] (2x - $x^{2}$) dx since 2x > $x^{2}$ on [0,2]

Q3HARD

Which is the integral of $x^{2}$ * e^x?

A) ($x^{2}$-2x+2)e^x + C
B) ($x^{2}$+2x+2)e^x + C
C) ($x^{2}$+2x-2)e^x + C
D) $x^{2}$*e^x + C
E) ($x^{2}$-2x)e^x + C
Show Answer

Answer: A — "($x^{2}$-2x+2)e^x + C" is correct because it applies integration by parts twice.

Q4HARD

Find ∫(1)/(x^2 + 4) dx

A) (1/2)arctan(x/2) + C
B) arctan(x/2) + C
C) (1/4)arctan(x/2) + C
D) (1/2)arctan(2x) + C
E) arctan(2x) + C
Show Answer

Answer: A — "(1/2)arctan(x/2) + C" is correct because it applies the formula for the integral of 1 / ($x^{2}$ + $a^{2}$) correctly.

Q5HARD

A population of bacteria grows at a rate of 200t^2, where t is time in hours. What is the total growth in the population from t = 0 to t = 5 hours?

A) ∫[0,5] 200t^2 dt = (200)(5) = 1000
B) ∫[0,5] 200t^2 dt = (200/2)($5^{2}$) = 2500
C) ∫[0,5] 200t^2 dt = 200($5^{2}$) = 5000
D) ∫[0,5] 200t^2 dt = (200/3)($5^{3}$) = 4166.67
E) ∫[0,5] 200t^2 dt = (200/3)($5^{2}$) = 1666.67
Show Answer

Answer: D — To find total growth, integrate the rate of growth function over the given time interval.

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Study Tips for Unit 3: Integrals

  • Focus on understanding concepts, not memorizing facts — CLEP tests application
  • Practice with timed questions to build exam-day speed
  • Review explanations for wrong answers — they reveal common misconceptions
  • Use flashcards for key terms, practice questions for deeper understanding

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