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.

  • A94𝜋4+1
  • B98𝜋
  • C32
  • D94𝜋
  • E92𝜋

Q2:

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

  • A𝜋332
  • B3𝜋3
  • C2𝜋
  • D232𝜋3
  • E𝜋

Q3:

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

  • A14𝜋33
  • B3𝜋4
  • C12𝜋+33
  • D14𝜋+33
  • E3𝜋2

Q4:

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

  • A3𝜋2
  • B2𝜋
  • C𝜋
  • D3𝜋
  • E𝜋4

Q5:

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

  • A94𝜋
  • B32𝜋
  • C92𝜋
  • D4+94𝜋
  • E2𝜋

Q6:

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

  • A2(2+𝜋)
  • B𝜋2
  • C4𝜋2
  • D2𝜋4
  • E4𝜋8

Q7:

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

  • A18𝜋273
  • B18𝜋2732
  • C15𝜋+7334
  • D3𝜋
  • E18𝜋+4532

Q8:

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

  • A4+𝜋2
  • B2
  • C4
  • D2+𝜋4
  • E2𝜋4

Q9:

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

  • A2(43𝜋)
  • B11𝜋24
  • C11𝜋
  • D24+11𝜋
  • E22𝜋

Q10:

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

  • A𝜋2
  • B32𝜋+4
  • C32𝜋
  • D3𝜋
  • E2𝜋

Q11:

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

  • A54𝜋
  • B2𝜋
  • C52𝜋
  • D𝜋
  • E74𝜋

Q12:

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

  • Aln42
  • B32𝜋
  • Cln4
  • D38𝜋
  • E34𝜋

Q13:

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

  • A𝜋3+33
  • B1+2𝜋3
  • C1+𝜋4
  • D3+2+𝜋3
  • E3+2+𝜋3

Q14:

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

  • A4𝜋
  • B4𝜋2
  • C4+𝜋2
  • D2+𝜋2
  • E2𝜋2

Q15:

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

  • A𝜋+332
  • B𝜋33
  • C2𝜋33
  • D2𝜋323
  • E𝜋332

Q16:

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

  • A𝜋4321
  • B2343𝜋
  • C𝜋
  • D323𝜋
  • E2𝜋

Q17:

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

  • A2𝜋3+23
  • B4𝜋323
  • C4𝜋3+23
  • D𝜋33
  • E8𝜋3+43

Q18:

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

  • A14
  • B3𝜋16
  • C𝜋8
  • D𝜋16
  • E12

Q19:

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

  • A313𝜋
  • B2𝜋
  • C2323𝜋
  • D23+43𝜋
  • E3+23𝜋

Q20:

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

  • A43
  • B2𝜋
  • C83
  • D4𝜋3
  • E8𝜋3

Q21:

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

  • A12
  • B𝜋2
  • C𝜋
  • D𝜋4
  • E2

Q22:

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

  • A4(4𝜋)
  • B4233+𝜋3
  • C234𝜋33
  • D434𝜋33
  • E4𝜋

Q23:

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

  • A83𝜋
  • B86
  • C2𝜋
  • D163𝜋
  • E43𝜋

Q24:

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

  • A12
  • B14
  • C𝜋16
  • D1
  • E𝜋8

Q25:

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

  • A3𝜋2
  • B𝜋
  • C3𝜋4
  • D𝜋2
  • E3𝜋

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