∫Calc Practice
Home›Calculus 3›Double integrals in polar coordinates

Double integrals in polar coordinates practice problems

Disks and circles: dA = r dr dθ. 12 problems with worked solutions; in 12 of them every equation is proved by a computer algebra system.

1 worked problems

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

Problem 11.33

medium
Evaluate \( \displaystyle \iint_D e^{- x^{2} - y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.
  1. In polar coordinates the disk is 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, and dA = r dr dθ.
    Reviewed
  2. \[ e^{- r^{2} \sin^{2}{\left(\theta \right)} - r^{2} \cos^{2}{\left(\theta \right)}} = e^{- r^{2}} \]
    The integrand in r and θ.✓ Proved
  3. \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2} r e^{- r^{2}}\, dr\, d\theta = - \frac{\pi}{e^{4}} + \pi \]
    Integrate, remembering the extra r.✓ Proved
Answer \( - \frac{\pi}{e^{4}} + \pi \)

✓ 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 in polar coordinates practice

Evaluate \( \displaystyle \iint_D \sqrt{x^{2} + y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
Problem 11.32easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D \sqrt{x^{2} + y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.
Problem 11.34easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} + 1 \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.
Problem 11.35easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.
Problem 11.37easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
Problem 11.38easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} + 1 \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.
Problem 11.39easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.
Problem 11.40easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D x^{2} + y^{2} + 1 \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
Problem 11.41easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D \sqrt{x^{2} + y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 4 \) using polar coordinates.
Problem 11.42easy✓ Every equation proved
Evaluate \( \displaystyle \iint_D e^{- x^{2} - y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
Problem 11.31medium✓ Every equation proved
Evaluate \( \displaystyle \iint_D e^{- x^{2} - y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 9 \) using polar coordinates.
Problem 11.36medium✓ Every equation proved