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Triple integrals practice problems

Integrate over a box, one variable at a time. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

5 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 11.43

easy
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 2] \).
  1. \[ \int\limits_{0}^{2} x^{2} z\, dz = 2 x^{2} \]
    Integrate in z.✓ Proved
  2. \[ \int\limits_{0}^{1} 2 x^{2}\, dy = 2 x^{2} \]
    Then y.✓ Proved
  3. \[ \int\limits_{0}^{2} 2 x^{2}\, dx = \frac{16}{3} \]
    Then x.✓ Proved
Answer \( \frac{16}{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 11.45

easy
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 1] \).
  1. \[ \int\limits_{0}^{1} 2 x z\, dz = x \]
    Integrate in z.✓ Proved
  2. \[ \int\limits_{0}^{3} x\, dy = 3 x \]
    Then y.✓ Proved
  3. \[ \int\limits_{0}^{1} 3 x\, dx = \frac{3}{2} \]
    Then x.✓ Proved
Answer \( \frac{3}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 11.48

easy
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 2] \).
  1. \[ \int\limits_{0}^{2} \left(x + y + z\right)\, dz = 2 x + 2 y + 2 \]
    Integrate in z.✓ Proved
  2. \[ \int\limits_{0}^{3} \left(2 x + 2 y + 2\right)\, dy = 6 x + 15 \]
    Then y.✓ Proved
  3. \[ \int\limits_{0}^{1} \left(6 x + 15\right)\, dx = 18 \]
    Then x.✓ Proved
Answer \( 18 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 11.50

easy
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 2] \).
  1. \[ \int\limits_{0}^{2} x y z\, dz = 2 x y \]
    Integrate in z.✓ Proved
  2. \[ \int\limits_{0}^{1} 2 x y\, dy = x \]
    Then y.✓ Proved
  3. \[ \int\limits_{0}^{2} x\, dx = 2 \]
    Then x.✓ Proved
Answer \( 2 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 11.90

easy
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 1] \).
  1. \[ \int\limits_{0}^{1} \left(x y + z\right)\, dz = x y + \frac{1}{2} \]
    Integrate in z.✓ Proved
  2. \[ \int\limits_{0}^{3} \left(x y + \frac{1}{2}\right)\, dy = \frac{9 x}{2} + \frac{3}{2} \]
    Then y.✓ Proved
  3. \[ \int\limits_{0}^{1} \left(\frac{9 x}{2} + \frac{3}{2}\right)\, dx = \frac{15}{4} \]
    Then x.✓ Proved
Answer \( \frac{15}{4} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More triple integrals practice

Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 1] \times [0, 3] \).
Problem 11.44easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 3] \).
Problem 11.46easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 1] \times [0, 2] \).
Problem 11.47easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 2] \times [0, 3] \).
Problem 11.49easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 3] \).
Problem 11.51easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 1] \).
Problem 11.52easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 1] \).
Problem 11.53easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 2] \).
Problem 11.54easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 3] \times [0, 3] \).
Problem 11.55easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 2] \times [0, 2] \).
Problem 11.56easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 3] \).
Problem 11.57easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 2] \).
Problem 11.88easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 3] \).
Problem 11.89easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 1] \).
Problem 11.91easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 3] \).
Problem 11.92easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 2] \times [0, 1] \).
Problem 11.93easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 1] \times [0, 2] \).
Problem 11.94easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 1] \times [0, 1] \).
Problem 11.95easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 1] \).
Problem 11.96easy✓ Every equation proved
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 1] \times [0, 1] \).
Problem 11.97easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 1] \).
Problem 11.98easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 1] \).
Problem 11.99easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 1] \times [0, 2] \).
Problem 11.100easy✓ Every equation proved
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 1] \times [0, 3] \).
Problem 11.101easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 2] \).
Problem 11.102easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 3] \).
Problem 11.119easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 2] \).
Problem 11.120easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 1] \).
Problem 11.121easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 3] \times [0, 3] \).
Problem 11.122easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 3] \).
Problem 11.123easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 3] \).
Problem 11.124easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 1] \times [0, 2] \).
Problem 11.125easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 1] \).
Problem 11.126easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 1] \).
Problem 11.142easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 2] \).
Problem 11.143easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 1] \).
Problem 11.144easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 1] \).
Problem 11.145easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 3] \).
Problem 11.146easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 2] \).
Problem 11.147easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 1] \).
Problem 11.148easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 3] \times [0, 1] \).
Problem 11.149easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 1] \).
Problem 11.160easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 3] \).
Problem 11.161easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 1] \times [0, 1] \).
Problem 11.162easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 3] \).
Problem 11.163easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 3] \times [0, 1] \).
Problem 11.164easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 2] \times [0, 1] \).
Problem 11.165easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 3] \).
Problem 11.166easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 1] \times [0, 3] \).
Problem 11.167easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 1] \times [0, 2] \).
Problem 11.169easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 3] \times [0, 2] \).
Problem 11.170easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 3] \).
Problem 11.171easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 1] \).
Problem 11.172easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 3] \times [0, 1] \).
Problem 11.173easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 2] \times [0, 3] \).
Problem 11.174easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 3] \).
Problem 11.175easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B 2 x z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 3] \times [0, 3] \).
Problem 11.176easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x + y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 1] \times [0, 3] \).
Problem 11.179easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 1] \times [0, 1] \).
Problem 11.180easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 1] \times [0, 3] \).
Problem 11.181easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 3] \).
Problem 11.182easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x^{2} z \, dV \) where \( \displaystyle B = [0, 1] \times [0, 2] \times [0, 2] \).
Problem 11.183easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 2] \times [0, 2] \times [0, 2] \).
Problem 11.184easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 3] \times [0, 2] \).
Problem 11.185easy✓ Nihil obstat
Evaluate \( \displaystyle \iiint_B x y + z \, dV \) where \( \displaystyle B = [0, 3] \times [0, 2] \times [0, 3] \).
Problem 11.186easy✓ Nihil obstat