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Double integrals over rectangles practice problems

Iterated integrals and Fubini's theorem. 56 problems with worked solutions; in 56 of them every equation is proved by a computer algebra system.

7 worked problems

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

Problem 11.1

medium
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 3] \times [0, 3] \).
  1. By Fubini's theorem, integrate in y first, then in x.
    Reviewed
  2. \[ \int\limits_{0}^{3} y e^{x}\, dy = \frac{9 e^{x}}{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{3} \frac{9 e^{x}}{2}\, dx = - \frac{9}{2} + \frac{9 e^{3}}{2} \]
    The outer integral.✓ Proved
Answer \( - \frac{9}{2} + \frac{9 e^{3}}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 11.2

easy
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 2] \).
  1. By Fubini's theorem, integrate in y first, then in x.
    Reviewed
  2. \[ \int\limits_{0}^{2} \left(x + 2 y\right)\, dy = 2 x + 4 \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} \left(2 x + 4\right)\, dx = 5 \]
    The outer integral.✓ Proved
Answer \( 5 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 11.3

easy
Evaluate \( \displaystyle \iint_R x \cos{\left(y \right)} \, dA \) where \( \displaystyle R = [0, 3] \times [0, \frac{\pi}{2}] \).
  1. By Fubini's theorem, integrate in y first, then in x.
    Reviewed
  2. \[ \int\limits_{0}^{\frac{\pi}{2}} x \cos{\left(y \right)}\, dy = x \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{3} x\, dx = \frac{9}{2} \]
    The outer integral.✓ Proved
Answer \( \frac{9}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 11.4

easy
Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 2] \).
  1. By Fubini's theorem, integrate in y first, then in x.
    Reviewed
  2. \[ \int\limits_{0}^{2} \left(x^{2} + y\right)\, dy = 2 x^{2} + 2 \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} \left(2 x^{2} + 2\right)\, dx = \frac{8}{3} \]
    The outer integral.✓ Proved
Answer \( \frac{8}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 11.5

easy
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 1] \).
  1. By Fubini's theorem, integrate in y first, then in x.
    Reviewed
  2. \[ \int\limits_{0}^{1} 6 x^{2} y\, dy = 3 x^{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 3 x^{2}\, dx = 8 \]
    The outer integral.✓ Proved
Answer \( 8 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 11.7

easy
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 1] \).
  1. By Fubini's theorem, integrate in y first, then in x.
    Reviewed
  2. \[ \int\limits_{0}^{1} x y\, dy = \frac{x}{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} \frac{x}{2}\, dx = 1 \]
    The outer integral.✓ Proved
Answer \( 1 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 11.9

easy
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 2] \times [0, 3] \).
  1. By Fubini's theorem, integrate in y first, then in x.
    Reviewed
  2. \[ \int\limits_{0}^{3} x y^{2}\, dy = 9 x \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 9 x\, dx = 18 \]
    The outer integral.✓ Proved
Answer \( 18 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. 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 double integrals over rectangles practice

Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 2] \).
Problem 11.6easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 2] \).
Problem 11.8easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 2] \).
Problem 11.10easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 1] \).
Problem 11.12easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 1] \).
Problem 11.13easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 1] \times [0, 3] \).
Problem 11.14easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 3] \times [0, 2] \).
Problem 11.58easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 1] \).
Problem 11.59easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 2] \).
Problem 11.60easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 1] \).
Problem 11.61easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 3] \).
Problem 11.62easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 3] \).
Problem 11.63easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 2] \).
Problem 11.64easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 3] \).
Problem 11.65easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x \cos{\left(y \right)} \, dA \) where \( \displaystyle R = [0, 1] \times [0, \frac{\pi}{2}] \).
Problem 11.66easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 3] \).
Problem 11.67easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 1] \).
Problem 11.68easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 3] \).
Problem 11.69easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 1] \).
Problem 11.70easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x \cos{\left(y \right)} \, dA \) where \( \displaystyle R = [0, 2] \times [0, \frac{\pi}{2}] \).
Problem 11.71easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 2] \).
Problem 11.72easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 3] \times [0, 3] \).
Problem 11.103easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 3] \).
Problem 11.104easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 3] \).
Problem 11.105easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 2] \times [0, 1] \).
Problem 11.106easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 1] \).
Problem 11.107easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 3] \).
Problem 11.108easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 2] \).
Problem 11.109easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 2] \times [0, 2] \).
Problem 11.127easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 2] \times [0, 2] \).
Problem 11.128easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 3] \times [0, 1] \).
Problem 11.130easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x^{2} + y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 3] \).
Problem 11.131easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 1] \times [0, 1] \).
Problem 11.132easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 2] \).
Problem 11.133easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 2] \).
Problem 11.134easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 3] \).
Problem 11.150easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 1] \).
Problem 11.151easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 3] \).
Problem 11.156easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x + 2 y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 1] \).
Problem 11.157easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R 6 x^{2} y \, dA \) where \( \displaystyle R = [0, 1] \times [0, 1] \).
Problem 11.177easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R x y^{2} \, dA \) where \( \displaystyle R = [0, 1] \times [0, 2] \).
Problem 11.178easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 3] \times [0, 2] \).
Problem 11.11medium✓ Nihil obstat
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 3] \times [0, 1] \).
Problem 11.15medium✓ Nihil obstat
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 2] \times [0, 2] \).
Problem 11.110medium✓ Nihil obstat
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 1] \times [0, 3] \).
Problem 11.129medium✓ Nihil obstat
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 2] \times [0, 1] \).
Problem 11.152medium✓ Nihil obstat
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 1] \times [0, 2] \).
Problem 11.153medium✓ Nihil obstat
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 2] \times [0, 3] \).
Problem 11.154medium✓ Nihil obstat
Evaluate \( \displaystyle \iint_R y e^{x} \, dA \) where \( \displaystyle R = [0, 1] \times [0, 1] \).
Problem 11.155medium✓ Nihil obstat