Worksheet: Area Bounded by Polar Curves

In this worksheet, we will practice calculating the area of the region enclosed by one or more polar curves.

Q1:

Find the area of the region enclosed by one petal of 𝑟=3(2𝜃)cos.

  • A 9 4 𝜋 4 + 1
  • B 9 8 𝜋
  • C 3 2
  • D 9 4 𝜋
  • E 9 2 𝜋

Q2:

Find the area of the region that lies inside the polar curve 𝑟=3𝜃cos but outside the polar curve 𝑟=1+𝜃cos.

  • A 𝜋 3 3 2
  • B 3 𝜋 3
  • C 2 𝜋
  • D 2 3 2 𝜋 3
  • E 𝜋

Q3:

Consider the polar curve 𝑟=12+𝜃cos. Find the area of the region inside its larger loop but outside its smaller loop.

  • A 1 4 𝜋 3 3
  • B 3 𝜋 4
  • C 1 2 𝜋 + 3 3
  • D 1 4 𝜋 + 3 3
  • E 3 𝜋 2

Q4:

Find the area of the region bounded by the polar curve 𝑟=1𝜃sin.

  • A 3 𝜋 2
  • B 2 𝜋
  • C 𝜋
  • D 3 𝜋
  • E 𝜋 4

Q5:

Find the area of the region below the polar axis and enclosed by 𝑟=2𝜃cos.

  • A 9 4 𝜋
  • B 3 2 𝜋
  • C 9 2 𝜋
  • D 4 + 9 4 𝜋
  • E 2 𝜋

Q6:

Find the area inside both 𝑟=2+2𝜃cos and 𝑟=2𝜃sin.

  • A 2 ( 2 + 𝜋 )
  • B 𝜋 2
  • C 4 𝜋 2
  • D 2 𝜋 4
  • E 4 𝜋 8

Q7:

Find the area of the region enclosed by the inner loop of 𝑟=3+6𝜃cos.

  • A 1 8 𝜋 2 7 3
  • B 1 8 𝜋 2 7 3 2
  • C 1 5 𝜋 + 7 3 3 4
  • D 3 𝜋
  • E 1 8 𝜋 + 4 5 3 2

Q8:

Find the area of the region that lies inside the polar curve 𝑟=1𝜃sin but outside the polar curve 𝑟=1.

  • A 4 + 𝜋 2
  • B2
  • C4
  • D 2 + 𝜋 4
  • E 2 𝜋 4

Q9:

Find the area of the region inside both 𝑟=32𝜃sin and 𝑟=3+2𝜃sin.

  • A 2 ( 4 3 𝜋 )
  • B 1 1 𝜋 2 4
  • C 1 1 𝜋
  • D 2 4 + 1 1 𝜋
  • E 2 2 𝜋

Q10:

Find the area of the region enclosed by 𝑟=1+𝜃sin.

  • A 𝜋 2
  • B 3 2 𝜋 + 4
  • C 3 2 𝜋
  • D 3 𝜋
  • E 2 𝜋

Q11:

Find the area of the region inside 𝑟=1+𝜃cos and outside 𝑟=𝜃cos.

  • A 5 4 𝜋
  • B 2 𝜋
  • C 5 2 𝜋
  • D 𝜋
  • E 7 4 𝜋

Q12:

Find the area of the region bounded by the polar curve 𝑟=1𝜃, where 𝜋2𝜃2𝜋.

  • A l n 4 2
  • B 3 2 𝜋
  • C l n 4
  • D 3 8 𝜋
  • E 3 4 𝜋

Q13:

Find the area of the region that lies inside both the polar curve 𝑟=22𝜃sin and the polar curve 𝑟=1.

  • A 𝜋 3 + 3 3
  • B 1 + 2 𝜋 3
  • C 1 + 𝜋 4
  • D 3 + 2 + 𝜋 3
  • E 3 + 2 + 𝜋 3

Q14:

Find the area enclosed by the loop of the right strophoid 𝑟=2𝜃𝜃cossec.

  • A 4 𝜋
  • B 4 𝜋 2
  • C 4 + 𝜋 2
  • D 2 + 𝜋 2
  • E 2 𝜋 2

Q15:

Find the area of the region enclosed by the inner loop of the polar curve 𝑟=1+2𝜃sin.

  • A 𝜋 + 3 3 2
  • B 𝜋 3 3
  • C 2 𝜋 3 3
  • D 2 𝜋 3 2 3
  • E 𝜋 3 3 2

Q16:

Find the area of the region that lies inside the circle 𝑟=3𝜃sin but outside the cardioid 𝑟=1+𝜃sin.

  • A 𝜋 4 3 2 1
  • B 2 3 4 3 𝜋
  • C 𝜋
  • D 3 2 3 𝜋
  • E 2 𝜋

Q17:

Find the area of the region that lies inside the polar curve 𝑟=4𝜃sin but outside the polar curve 𝑟=2.

  • A 2 𝜋 3 + 2 3
  • B 4 𝜋 3 2 3
  • C 4 𝜋 3 + 2 3
  • D 𝜋 3 3
  • E 8 𝜋 3 + 4 3

Q18:

Find the area of the region enclosed by one loop of the polar curve 𝑟=4𝜃sin.

  • A 1 4
  • B 3 𝜋 1 6
  • C 𝜋 8
  • D 𝜋 1 6
  • E 1 2

Q19:

Find the area inside the circle 𝑟=4𝜃cos and outside the circle 𝑟=2.

  • A 3 1 3 𝜋
  • B 2 𝜋
  • C 2 3 2 3 𝜋
  • D 2 3 + 4 3 𝜋
  • E 3 + 2 3 𝜋

Q20:

Find the area of the region enclosed by one loop of the polar curve 𝑟=43𝜃cos.

  • A 4 3
  • B 2 𝜋
  • C 8 3
  • D 4 𝜋 3
  • E 8 𝜋 3

Q21:

Find the area of the region bounded by the polar curve 𝑟=𝜃+𝜃sincos, where 0𝜃𝜋.

  • A 1 2
  • B 𝜋 2
  • C 𝜋
  • D 𝜋 4
  • E2

Q22:

Find the area of the region common to the interior of 𝑟=4(2𝜃)sin and 𝑟=2.

  • A 4 ( 4 𝜋 )
  • B 4 2 3 3 + 𝜋 3
  • C 2 3 4 𝜋 3 3
  • D 4 3 4 𝜋 3 3
  • E 4 𝜋

Q23:

Find the area of the region enclosed by one petal of 𝑟=4(3𝜃)cos.

  • A 8 3 𝜋
  • B 8 6
  • C 2 𝜋
  • D 1 6 3 𝜋
  • E 4 3 𝜋

Q24:

Find the area enclosed by one loop of the rose with polar equation 𝑟=2𝜃cos.

  • A 1 2
  • B 1 4
  • C 𝜋 1 6
  • D1
  • E 𝜋 8

Q25:

Find the area of the region enclosed by the polar curve 𝑟=1+(5𝜃)cos.

  • A 3 𝜋 2
  • B 𝜋
  • C 3 𝜋 4
  • D 𝜋 2
  • E 3 𝜋

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