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Double integrals over general regions practice problems

Type I and Type II regions: set the bounds, then integrate. 48 problems with worked solutions; in 48 of them every equation is proved by a computer algebra system.

20 worked problems

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

Problem 11.26

medium
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to sqrt(x).
    Reviewed
  2. \[ \int\limits_{0}^{\sqrt{x}} x^{2}\, dy = x^{\frac{5}{2}} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} x^{\frac{5}{2}}\, dx = \frac{16 \sqrt{2}}{7} \]
    The outer integral.✓ Proved
Answer \( \frac{16 \sqrt{2}}{7} \)

✓ 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 11.86

medium
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to sqrt(x).
    Reviewed
  2. \[ \int\limits_{0}^{\sqrt{x}} \left(x + y\right)\, dy = x^{\frac{3}{2}} + \frac{x}{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} \left(x^{\frac{3}{2}} + \frac{x}{2}\right)\, dx = 1 + \frac{8 \sqrt{2}}{5} \]
    The outer integral.✓ Proved
Answer \( 1 + \frac{8 \sqrt{2}}{5} \)

✓ 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 11.159

medium
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to sqrt(x).
    Reviewed
  2. \[ \int\limits_{0}^{\sqrt{x}} 2 x\, dy = 2 x^{\frac{3}{2}} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 2 x^{\frac{3}{2}}\, dx = \frac{16 \sqrt{2}}{5} \]
    The outer integral.✓ Proved
Answer \( \frac{16 \sqrt{2}}{5} \)

✓ 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 11.16

easy
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to x.
    Reviewed
  2. \[ \int\limits_{0}^{x} x^{2}\, dy = x^{3} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} x^{3}\, dx = \frac{1}{4} \]
    The outer integral.✓ Proved
Answer \( \frac{1}{4} \)

✓ 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 11.17

easy
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to x.
    Reviewed
  2. \[ \int\limits_{0}^{x} \left(x + y\right)\, dy = \frac{3 x^{2}}{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} \frac{3 x^{2}}{2}\, dx = 4 \]
    The outer integral.✓ Proved
Answer \( 4 \)

✓ 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 11.18

easy
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to 2*x.
    Reviewed
  2. \[ \int\limits_{0}^{2 x} 2 x\, dy = 4 x^{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} 4 x^{2}\, dx = \frac{4}{3} \]
    The outer integral.✓ Proved
Answer \( \frac{4}{3} \)

✓ 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 11.19

easy
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to x.
    Reviewed
  2. \[ \int\limits_{0}^{x} x y\, dy = \frac{x^{3}}{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} \frac{x^{3}}{2}\, dx = \frac{1}{8} \]
    The outer integral.✓ Proved
Answer \( \frac{1}{8} \)

✓ 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 11.20

easy
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to x**2.
    Reviewed
  2. \[ \int\limits_{0}^{x^{2}} 2 x\, dy = 2 x^{3} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 2 x^{3}\, dx = 8 \]
    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 11.21

easy
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to x**2.
    Reviewed
  2. \[ \int\limits_{0}^{x^{2}} 6 x y\, dy = 3 x^{5} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 3 x^{5}\, dx = 32 \]
    The outer integral.✓ Proved
Answer \( 32 \)

✓ 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 11.23

easy
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to 2*x.
    Reviewed
  2. \[ \int\limits_{0}^{2 x} x y\, dy = 2 x^{3} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 2 x^{3}\, dx = 8 \]
    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 11.24

easy
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to sqrt(x).
    Reviewed
  2. \[ \int\limits_{0}^{\sqrt{x}} x y\, dy = \frac{x^{2}}{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} \frac{x^{2}}{2}\, dx = \frac{1}{6} \]
    The outer integral.✓ Proved
Answer \( \frac{1}{6} \)

✓ 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 11.27

easy
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to x.
    Reviewed
  2. \[ \int\limits_{0}^{x} 6 x y\, dy = 3 x^{3} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} 3 x^{3}\, dx = \frac{3}{4} \]
    The outer integral.✓ Proved
Answer \( \frac{3}{4} \)

✓ 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 11.28

easy
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to x.
    Reviewed
  2. \[ \int\limits_{0}^{x} 2 x\, dy = 2 x^{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 2 x^{2}\, dx = \frac{16}{3} \]
    The outer integral.✓ Proved
Answer \( \frac{16}{3} \)

✓ 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 11.29

easy
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to 2*x.
    Reviewed
  2. \[ \int\limits_{0}^{2 x} x^{2}\, dy = 2 x^{3} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} 2 x^{3}\, dx = \frac{1}{2} \]
    The outer integral.✓ Proved
Answer \( \frac{1}{2} \)

✓ 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 11.73

easy
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to x.
    Reviewed
  2. \[ \int\limits_{0}^{x} y\, dy = \frac{x^{2}}{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} \frac{x^{2}}{2}\, dx = \frac{1}{6} \]
    The outer integral.✓ Proved
Answer \( \frac{1}{6} \)

✓ 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 11.76

easy
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to 2*x.
    Reviewed
  2. \[ \int\limits_{0}^{2 x} 6 x y\, dy = 12 x^{3} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 12 x^{3}\, dx = 48 \]
    The outer integral.✓ Proved
Answer \( 48 \)

✓ 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 11.77

easy
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to sqrt(x).
    Reviewed
  2. \[ \int\limits_{0}^{\sqrt{x}} 6 x y\, dy = 3 x^{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} 3 x^{2}\, dx = 1 \]
    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 11.79

easy
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
  1. D is Type I: for each x in [0, 1], y runs from 0 to x**2.
    Reviewed
  2. \[ \int\limits_{0}^{x^{2}} x y\, dy = \frac{x^{5}}{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{1} \frac{x^{5}}{2}\, dx = \frac{1}{12} \]
    The outer integral.✓ Proved
Answer \( \frac{1}{12} \)

✓ 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 11.81

easy
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to sqrt(x).
    Reviewed
  2. \[ \int\limits_{0}^{\sqrt{x}} y\, dy = \frac{x}{2} \]
    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 11.85

easy
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
  1. D is Type I: for each x in [0, 2], y runs from 0 to 2*x.
    Reviewed
  2. \[ \int\limits_{0}^{2 x} y\, dy = 2 x^{2} \]
    The inner integral.✓ Proved
  3. \[ \int\limits_{0}^{2} 2 x^{2}\, dx = \frac{16}{3} \]
    The outer integral.✓ Proved
Answer \( \frac{16}{3} \)

✓ 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

More double integrals over general regions practice

Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.22easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.25easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.30easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.74easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.75easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.78easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.80easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.82easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.83easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.84easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.87easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.111easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.112easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.113easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.114easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.115easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.116easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.117easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.118easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.135easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.136easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.137easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.138easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x^{2} \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.139easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D x + y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = 2 x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.140easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.141easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x^{2} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
Problem 11.158easy✓ Nihil obstat
Evaluate \( \displaystyle \iint_D 2 x \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = \sqrt{x} \), \( \displaystyle x = 0 \) and \( \displaystyle x = 1 \).
Problem 11.168easy✓ Nihil obstat