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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.

Find the area of the region swept by \( \displaystyle r = \theta \) for \( \displaystyle 0 \le \theta \le \pi \).

Problem 8.16easy✓ Every equation proved

Find the area of the region inside \( \displaystyle r = 6 \sin{\left(\theta \right)} \).

Problem 8.17easy✓ Every equation proved

Find the area of the region swept by \( \displaystyle r = \theta \) for \( \displaystyle 0 \le \theta \le 2 \pi \).

Problem 8.18easy✓ Every equation proved

Find the area of the region inside the cardioid \( \displaystyle r = 4 \cos{\left(\theta \right)} + 4 \).

Problem 8.19easy✓ Every equation proved

Find the area of one petal of the rose \( \displaystyle r = \cos{\left(3 \theta \right)} \).

Problem 8.20easy✓ Every equation proved

Find the area of one petal of the rose \( \displaystyle r = 3 \cos{\left(2 \theta \right)} \).

Problem 8.21easy✓ Every equation proved

Find the area of one petal of the rose \( \displaystyle r = 4 \cos{\left(2 \theta \right)} \).

Problem 8.22easy✓ Every equation proved

Find the area of the region inside \( \displaystyle r = \sin{\left(\theta \right)} \).

Problem 8.23easy✓ Every equation proved

Find the area of the region inside the cardioid \( \displaystyle r = 2 \cos{\left(\theta \right)} + 2 \).

Problem 8.24easy✓ Every equation proved

Find the area of one petal of the rose \( \displaystyle r = 2 \cos{\left(2 \theta \right)} \).

Problem 8.25easy✓ Every equation proved

Find the area of the region inside \( \displaystyle r = 5 \sin{\left(\theta \right)} \).

Problem 8.26easy✓ Every equation proved

Find the area of the region inside the cardioid \( \displaystyle r = 3 \cos{\left(\theta \right)} + 3 \).

Problem 8.27easy✓ Every equation proved

Find the area of one petal of the rose \( \displaystyle r = \cos{\left(2 \theta \right)} \).

Problem 8.28easy✓ Every equation proved

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✓ Every equation proved

Find the area of the region inside the cardioid \( \displaystyle r = \cos{\left(\theta \right)} + 1 \).

Problem 8.47easy✓ Every equation proved

Find the area of one petal of the rose \( \displaystyle r = 3 \cos{\left(3 \theta \right)} \).

Problem 8.48easy✓ Every equation proved

Find the area of the region inside \( \displaystyle r = 3 \sin{\left(\theta \right)} \).

Problem 8.49easy✓ Every equation proved

Find the area of one petal of the rose \( \displaystyle r = 4 \cos{\left(3 \theta \right)} \).

Problem 8.50easy✓ Every equation proved