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Definite Integrals

13 practice questions with full worked solutions. Free, no account needed.

Problems & worked solutions

Problem #0201 International

Problem 1 — Definite Integrals

∫01x dx\displaystyle\int_0^1 x \, dx equals:

Show answer & worked solution
  1. A. 00
  2. B. 12\dfrac{1}{2}✓ correct
  3. C. 11
  4. D. 22

F(x)=x22F(x) = \dfrac{x^2}{2}. ∫01x dx=F(1)−F(0)=12\int_0^1 x \, dx = F(1) - F(0) = \dfrac{1}{2}.

Problem #0202 International

Problem 2 — Definite Integrals

∫12x2 dx\displaystyle\int_1^2 x^2 \, dx equals:

Show answer & worked solution
  1. A. 13\dfrac{1}{3}
  2. B. 73\dfrac{7}{3}✓ correct
  3. C. 83\dfrac{8}{3}
  4. D. 33

F(x)=x33F(x) = \dfrac{x^3}{3}. F(2)−F(1)=83−13=73F(2) - F(1) = \dfrac{8}{3} - \dfrac{1}{3} = \dfrac{7}{3}.

Problem #0203 International

Problem 3 — Definite Integrals

∫−11x3 dx\displaystyle\int_{-1}^{1} x^3 \, dx equals:

Show answer & worked solution
  1. A. 00✓ correct
  2. B. 14\dfrac{1}{4}
  3. C. 12\dfrac{1}{2}
  4. D. 11

For an odd function ff and symmetric interval [−a,a][-a, a]: ∫−aaf=0\int_{-a}^{a} f = 0.

Problem #0206 International

Problem 4 — Definite Integrals

The area under the curve y=x2y = x^2 between x=0x = 0 and x=3x = 3 equals:

Show answer & worked solution
  1. A. 33
  2. B. 66
  3. C. 99✓ correct
  4. D. 2727

∫03x2 dx=273=9\int_0^3 x^2 \, dx = \dfrac{27}{3} = 9.

Problem #0205 International

Problem 5 — Definite Integrals

∫01ex dx\displaystyle\int_0^1 e^x \, dx equals:

Show answer & worked solution
  1. A. 11
  2. B. ee
  3. C. e−1e - 1✓ correct
  4. D. e+1e + 1

∫01ex dx=e1−e0=e−1\int_0^1 e^x \, dx = e^1 - e^0 = e - 1.

Problem #0208 International

Problem 6 — Definite Integrals

The average value of f(x)=x2f(x) = x^2 on [0,3][0, 3] is:

Show answer & worked solution
  1. A. 11
  2. B. 32\dfrac{3}{2}
  3. C. 33✓ correct
  4. D. 99

13∫03x2 dx=13⋅9=3\dfrac{1}{3}\int_0^3 x^2 \, dx = \dfrac{1}{3} \cdot 9 = 3.

Problem #0207 International

Problem 7 — Definite Integrals

∫012x(x2+1)3 dx\displaystyle\int_0^1 2x(x^2 + 1)^3 \, dx equals:

Show answer & worked solution
  1. A. 11
  2. B. 154\dfrac{15}{4}✓ correct
  3. C. 174\dfrac{17}{4}
  4. D. 44

u=x2+1⇒du=2x dxu = x^2 + 1 \Rightarrow du = 2x \, dx. When x=0x = 0, u=1u = 1; when x=1x = 1, u=2u = 2. ∫12u3 du=16−14=154\int_1^2 u^3 \, du = \dfrac{16 - 1}{4} = \dfrac{15}{4}.

Problem #0204 International

Problem 8 — Definite Integrals

∫0π/2cos⁡x dx\displaystyle\int_0^{\pi/2} \cos x \, dx equals:

Show answer & worked solution
  1. A. 00
  2. B. 12\dfrac{1}{2}
  3. C. 11✓ correct
  4. D. π2\dfrac{\pi}{2}

F(x)=sin⁡xF(x) = \sin x. F(π/2)−F(0)=1−0=1F(\pi/2) - F(0) = 1 - 0 = 1.

Problem #0903 US AP

Problem 9 — Definite Integrals

The region RR is bounded by the graph of f(x)=3x2+2f(x) = 3x^2 + 2, the xx-axis, and the vertical lines x=1x = 1 and x=3x = 3. What is the area of RR?

Show answer & worked solution
  1. A. 3030✓ correct
  2. B. 3333
  3. C. 2626
  4. D. 3232

Area=∫13(3x2+2) dx\text{Area} = \int_1^3 \left(3x^2 + 2\right)\,dx

∫(3x2+2) dx=x3+2x\int \left(3x^2 + 2\right)\,dx = x^3 + 2x

[x3+2x]13=(27+6)−(1+2)\Big[x^3 + 2x\Big]_1^3 = (27 + 6) - (1 + 2)

=33−3=30= 33 - 3 = 30

Problem #0209 International

Problem 10 — Definite Integrals

∫1eln⁡x dx\displaystyle\int_1^e \ln x \, dx equals:

Show answer & worked solution
  1. A. 11✓ correct
  2. B. ee
  3. C. e−1e - 1
  4. D. 00

∫ln⁡x dx=xln⁡x−x+C\int \ln x \, dx = x \ln x - x + C. Evaluate: (e⋅1−e)−(1⋅0−1)=0−(−1)=1(e \cdot 1 - e) - (1 \cdot 0 - 1) = 0 - (-1) = 1.

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