Worksheet: Area Bounded by Polar Curves

In this worksheet, we will practice calculating the area of the region enclosed by a polar curve and finding the area of a region bounded by two polar curves.

Q1:

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

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

Q2:

Find the area of the region that lies inside the polar curve 𝑟 = 3 𝜃 c o s but outside the polar curve 𝑟 = 1 + 𝜃 c o s .

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

Q3:

Consider the polar curve 𝑟 = 1 2 + 𝜃 c o s . Find the area of the region inside its larger loop but outside its smaller loop.

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

Q4:

Find the area of the region bounded by the polar curve 𝑟 = 1 𝜃 s i n .

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

Q5:

Find the area of the region below the polar axis and enclosed by 𝑟 = 2 𝜃 c o s .

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

Q6:

Find the area inside both 𝑟 = 2 + 2 𝜃 c o s and 𝑟 = 2 𝜃 s i n .

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

Q7:

Find the area of the region enclosed by the inner loop of 𝑟 = 3 + 6 𝜃 c o s .

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

Q8:

Find the area of the region that lies inside the polar curve 𝑟 = 1 𝜃 s i n but outside the polar curve 𝑟 = 1 .

  • A2
  • B 4 + 𝜋 2
  • C4
  • D 2 + 𝜋 4
  • E 2 𝜋 4

Q9:

Find the area of the region inside both 𝑟 = 3 2 𝜃 s i n and 𝑟 = 3 + 2 𝜃 s i n .

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

Q10:

Find the area of the region enclosed by 𝑟 = 1 + 𝜃 s i n .

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

Q11:

Find the area of the region inside 𝑟 = 1 + 𝜃 c o s and outside 𝑟 = 𝜃 c o s .

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

Q12:

Find the area of the region bounded by the polar curve 𝑟 = 1 𝜃 , where 𝜋 2 𝜃 2 𝜋 .

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

Q13:

Find the area of the region that lies inside both the polar curve 𝑟 = 2 2 𝜃 2 s i n and the polar curve 𝑟 = 1 .

  • A 1 + 2 𝜋 3
  • B 𝜋 3 + 3 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 = 2 𝜃 𝜃 c o s s e c .

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

Q15:

Find the area of the region enclosed by the inner loop of the polar curve 𝑟 = 1 + 2 𝜃 s i n .

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

Q16:

Find the area of the region that lies inside the circle 𝑟 = 3 𝜃 s i n but outside the cardioid 𝑟 = 1 + 𝜃 s i n .

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

Q17:

Find the area of the region that lies inside the polar curve 𝑟 = 4 𝜃 s i n but outside the polar curve 𝑟 = 2 .

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

Q18:

Find the area of the region enclosed by one loop of the polar curve 𝑟 = 4 𝜃 s i n .

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

Q19:

Find the area inside the circle 𝑟 = 4 𝜃 c o s and outside the circle 𝑟 = 2 .

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

Q20:

Find the area of the region enclosed by one loop of the polar curve 𝑟 = 4 3 𝜃 c o s .

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

Q21:

Find the area of the region bounded by the polar curve 𝑟 = 𝜃 + 𝜃 s i n c o s , where 0 𝜃 𝜋 .

  • A 𝜋 4
  • B 1 2
  • C2
  • D 𝜋 2
  • E 𝜋

Q22:

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

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

Q23:

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

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

Q24:

Find the area enclosed by one loop of the rose with polar equation 𝑟 = 2 𝜃 c o s .

  • A 1 2
  • B 𝜋 1 6
  • C 1 4
  • D 𝜋 8
  • E1

Q25:

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

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

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