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Area in polar coordinates practice problems

½∫ r² dθ: petals, cardioids and circles. 20 problems with worked solutions; in 20 of them every equation is proved by a computer algebra system.

8 worked problems

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

Problem 8.16

easy
Find the area of the region swept by \( \displaystyle r = \theta \) for \( \displaystyle 0 \le \theta \le \pi \).
  1. Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
    Reviewed
  2. Here θ runs from 0 to pi.
    Reviewed
  3. \[ \int\limits_{0}^{\pi} \frac{\theta^{2}}{2}\, d\theta = \frac{\pi^{3}}{6} \]
    Integrate.✓ Proved
Answer \( \frac{\pi^{3}}{6} \)

✓ Nihil obstat Lines: 1 proved, 2 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 8.17

easy
Find the area of the region inside \( \displaystyle r = 6 \sin{\left(\theta \right)} \).
  1. Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
    Reviewed
  2. Here θ runs from 0 to pi.
    Reviewed
  3. \[ \int\limits_{0}^{\pi} 18 \sin^{2}{\left(\theta \right)}\, d\theta = 9 \pi \]
    Integrate.✓ Proved
Answer \( 9 \pi \)

✓ Nihil obstat Lines: 1 proved, 2 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 8.18

easy
Find the area of the region swept by \( \displaystyle r = \theta \) for \( \displaystyle 0 \le \theta \le 2 \pi \).
  1. Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
    Reviewed
  2. Here θ runs from 0 to 2*pi.
    Reviewed
  3. \[ \int\limits_{0}^{2 \pi} \frac{\theta^{2}}{2}\, d\theta = \frac{4 \pi^{3}}{3} \]
    Integrate.✓ Proved
Answer \( \frac{4 \pi^{3}}{3} \)

✓ Nihil obstat Lines: 1 proved, 2 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 8.19

easy
Find the area of the region inside the cardioid \( \displaystyle r = 4 \cos{\left(\theta \right)} + 4 \).
  1. Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
    Reviewed
  2. Here θ runs from 0 to 2*pi.
    Reviewed
  3. \[ \int\limits_{0}^{2 \pi} \frac{\left(4 \cos{\left(\theta \right)} + 4\right)^{2}}{2}\, d\theta = 24 \pi \]
    Integrate.✓ Proved
Answer \( 24 \pi \)

✓ Nihil obstat Lines: 1 proved, 2 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 8.20

easy
Find the area of one petal of the rose \( \displaystyle r = \cos{\left(3 \theta \right)} \).
  1. Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
    Reviewed
  2. Here θ runs from -pi/6 to pi/6.
    Reviewed
  3. \[ \int\limits_{- \frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\cos^{2}{\left(3 \theta \right)}}{2}\, d\theta = \frac{\pi}{12} \]
    Integrate.✓ Proved
Answer \( \frac{\pi}{12} \)

✓ Nihil obstat Lines: 1 proved, 2 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 8.21

easy
Find the area of one petal of the rose \( \displaystyle r = 3 \cos{\left(2 \theta \right)} \).
  1. Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
    Reviewed
  2. Here θ runs from -pi/4 to pi/4.
    Reviewed
  3. \[ \int\limits_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \frac{9 \cos^{2}{\left(2 \theta \right)}}{2}\, d\theta = \frac{9 \pi}{8} \]
    Integrate.✓ Proved
Answer \( \frac{9 \pi}{8} \)

✓ Nihil obstat Lines: 1 proved, 2 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 8.23

easy
Find the area of the region inside \( \displaystyle r = \sin{\left(\theta \right)} \).
  1. Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
    Reviewed
  2. Here θ runs from 0 to pi.
    Reviewed
  3. \[ \int\limits_{0}^{\pi} \frac{\sin^{2}{\left(\theta \right)}}{2}\, d\theta = \frac{\pi}{4} \]
    Integrate.✓ Proved
Answer \( \frac{\pi}{4} \)

✓ Nihil obstat Lines: 1 proved, 2 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 8.47

easy
Find the area of the region inside the cardioid \( \displaystyle r = \cos{\left(\theta \right)} + 1 \).
  1. Area in polar coordinates is ½∫ r² dθ over the angles that trace the region once.
    Reviewed
  2. Here θ runs from 0 to 2*pi.
    Reviewed
  3. \[ \int\limits_{0}^{2 \pi} \frac{\left(\cos{\left(\theta \right)} + 1\right)^{2}}{2}\, d\theta = \frac{3 \pi}{2} \]
    Integrate.✓ Proved
Answer \( \frac{3 \pi}{2} \)

✓ Nihil obstat Lines: 1 proved, 2 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 area in polar coordinates practice

Find the area of one petal of the rose \( \displaystyle r = 4 \cos{\left(2 \theta \right)} \).
Problem 8.22easy✓ Nihil obstat
Find the area of the region inside the cardioid \( \displaystyle r = 2 \cos{\left(\theta \right)} + 2 \).
Problem 8.24easy✓ Nihil obstat
Find the area of one petal of the rose \( \displaystyle r = 2 \cos{\left(2 \theta \right)} \).
Problem 8.25easy✓ Nihil obstat
Find the area of the region inside \( \displaystyle r = 5 \sin{\left(\theta \right)} \).
Problem 8.26easy✓ Nihil obstat
Find the area of the region inside the cardioid \( \displaystyle r = 3 \cos{\left(\theta \right)} + 3 \).
Problem 8.27easy✓ Nihil obstat
Find the area of one petal of the rose \( \displaystyle r = \cos{\left(2 \theta \right)} \).
Problem 8.28easy✓ Nihil obstat
Find the area of one petal of the rose \( \displaystyle r = 2 \cos{\left(3 \theta \right)} \).
Problem 8.29easy✓ Every equation proved
Find the area of the region inside \( \displaystyle r = 2 \sin{\left(\theta \right)} \).
Problem 8.30easy✓ Every equation proved
Find the area of the region inside \( \displaystyle r = 4 \sin{\left(\theta \right)} \).
Problem 8.46easy✓ Nihil obstat
Find the area of one petal of the rose \( \displaystyle r = 3 \cos{\left(3 \theta \right)} \).
Problem 8.48easy✓ Nihil obstat
Find the area of the region inside \( \displaystyle r = 3 \sin{\left(\theta \right)} \).
Problem 8.49easy✓ Nihil obstat
Find the area of one petal of the rose \( \displaystyle r = 4 \cos{\left(3 \theta \right)} \).
Problem 8.50easy✓ Nihil obstat