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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.

46

Practice Questions

28

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 46.

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 = x2x^{2} and y = 2x on [0,2]

A.

∫[0,2] (x2x^{2} - 2x) dx

B.

∫[0,2] (2x - x2x^{2}) dx

C.

∫[0,2] (x2x^{2} + 2x) dx

D.

∫[0,2] (2x + x2x^{2}) dx

E.

∫[0,2] (x2x^{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 x2x^{2} * e^x?

A.

(x2x^{2}-2x+2)e^x + C

B.

(x2x^{2}+2x+2)e^x + C

C.

(x2x^{2}+2x-2)e^x + C

D.

x2x^{2}*e^x + C

E.

(x2x^{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 ∫1x2+4\frac{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 200t2200t^{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] 200t2200t^{2} dt = (200)(5) = 1000

B.

∫[0,5] 200t^2 dt = (200/2)(525^{2}) = 2500

C.

∫[0,5] 200t^2 dt = 200(525^{2}) = 5000

D.

∫[0,5] 200t^2 dt = (200/3)(535^{3}) = 4166.67

E.

∫[0,5] 200t^2 dt = (200/3)(525^{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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