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Lesson: Area Bounded by Polar Curves

Worksheet • 25 Questions

Q1:

Find the area of the region enclosed by one petal of π‘Ÿ = 3 ( 2 πœƒ ) c o s .

  • A 9 8 πœ‹
  • B 9 4 ο€» πœ‹ 4 + 1 
  • C 9 2 πœ‹
  • D 3 2
  • E 9 4 πœ‹

Q2:

Find the area of the region that lies inside the polar curve π‘Ÿ = 3 πœƒ c o s but outside the polar curve π‘Ÿ = 1 + πœƒ c o s .

  • A πœ‹
  • B πœ‹ βˆ’ 3 √ 3 2
  • C 2 πœ‹
  • D 2 √ 3 βˆ’ 2 πœ‹ 3
  • E √ 3 βˆ’ πœ‹ 3

Q3:

Consider the polar curve π‘Ÿ = 1 2 + πœƒ c o s . Find the area of the region inside its larger loop but outside its smaller loop.

  • A 1 4 ο€» πœ‹ + 3 √ 3 
  • B 3 πœ‹ 2
  • C 1 2 ο€» πœ‹ + 3 √ 3 
  • D 3 πœ‹ 4
  • E 1 4 ο€» πœ‹ βˆ’ 3 √ 3 

Q4:

Find the area of the region bounded by the polar curve π‘Ÿ = 1 βˆ’ πœƒ s i n .

  • A 3 πœ‹ 2
  • B πœ‹ 4
  • C 3 πœ‹
  • D 2 πœ‹
  • E πœ‹

Q5:

Find the area of the region below the polar axis and enclosed by π‘Ÿ = 2 βˆ’ πœƒ c o s .

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

Q6:

Find the area inside both π‘Ÿ = 2 + 2 πœƒ c o s and π‘Ÿ = 2 πœƒ s i n .

  • A 2 πœ‹ βˆ’ 4
  • B 2 ( 2 + πœ‹ )
  • C 4 πœ‹ βˆ’ 8
  • D πœ‹ 2
  • E 4 πœ‹ βˆ’ 2

Q7:

Find the area of the region enclosed by the inner loop of π‘Ÿ = 3 + 6 πœƒ c o s .

  • A 1 8 πœ‹ βˆ’ 2 7 √ 3 2
  • B 1 5 πœ‹ + 7 3 √ 3 4
  • C 1 8 πœ‹ βˆ’ 2 7 √ 3
  • D 3 πœ‹
  • E 1 8 πœ‹ + 4 5 √ 3 2

Q8:

Find the area of the region that lies inside the polar curve π‘Ÿ = 1 βˆ’ πœƒ s i n but outside the polar curve π‘Ÿ = 1 .

  • A 2 + πœ‹ 4
  • B 2 βˆ’ πœ‹ 4
  • C 4 + πœ‹ 2
  • D4
  • E2

Q9:

Find the area of the region inside both π‘Ÿ = 3 βˆ’ 2 πœƒ s i n and π‘Ÿ = βˆ’ 3 + 2 πœƒ s i n .

  • A 1 1 πœ‹ βˆ’ 2 4
  • B 1 1 πœ‹
  • C 2 4 + 1 1 πœ‹
  • D 2 ( 4 βˆ’ 3 πœ‹ )
  • E 2 2 πœ‹

Q10:

Find the area of the region enclosed by π‘Ÿ = 1 + πœƒ s i n .

  • A 3 2 πœ‹
  • B 3 2 πœ‹ + 4
  • C 2 πœ‹
  • D πœ‹ 2
  • E 3 πœ‹

Q11:

Find the area of the region inside π‘Ÿ = 1 + πœƒ c o s and outside π‘Ÿ = πœƒ c o s .

  • A 5 4 πœ‹
  • B πœ‹
  • C 5 2 πœ‹
  • D 2 πœ‹
  • E 7 4 πœ‹

Q12:

Find the area of the region bounded by the polar curve π‘Ÿ = 1 πœƒ , where πœ‹ 2 ≀ πœƒ ≀ 2 πœ‹ .

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

Q13:

Find the area of the region that lies inside both the polar curve π‘Ÿ = 2 2 πœƒ 2 s i n and the polar curve π‘Ÿ = 1 .

  • A βˆ’ √ 3 + 2 + πœ‹ 3
  • B √ 3 + 2 + πœ‹ 3
  • C βˆ’ πœ‹ 3 + 3 √ 3
  • D 1 + πœ‹ 4
  • E 1 + 2 πœ‹ 3

Q14:

Find the area enclosed by the loop of the right strophoid π‘Ÿ = 2 πœƒ βˆ’ πœƒ c o s s e c .

  • A 2 βˆ’ πœ‹ 2
  • B 4 βˆ’ πœ‹
  • C 4 βˆ’ πœ‹ 2
  • D 2 + πœ‹ 2
  • E 4 + πœ‹ 2

Q15:

Find the area of the region enclosed by the inner loop of the polar curve π‘Ÿ = 1 + 2 πœƒ s i n .

  • A πœ‹ βˆ’ 3 √ 3 2
  • B πœ‹ + 3 √ 3 2
  • C 2 πœ‹ βˆ’ 3 √ 3
  • D 2 πœ‹ 3 βˆ’ 2 √ 3
  • E πœ‹ 3 βˆ’ √ 3

Q16:

Find the area of the region that lies inside the circle π‘Ÿ = 3 πœƒ s i n but outside the cardioid π‘Ÿ = 1 + πœƒ s i n .

  • A πœ‹
  • B 2 √ 3 βˆ’ 4 3 πœ‹
  • C πœ‹ 4 βˆ’ √ 3 2 βˆ’ 1
  • D √ 3 βˆ’ 2 3 πœ‹
  • E 2 πœ‹

Q17:

Find the area of the region that lies inside the polar curve π‘Ÿ = 4 πœƒ s i n but outside the polar curve π‘Ÿ = 2 .

  • A 4 πœ‹ 3 + 2 √ 3
  • B 4 πœ‹ 3 βˆ’ 2 √ 3
  • C 8 πœ‹ 3 + 4 √ 3
  • D βˆ’ 2 πœ‹ 3 + 2 √ 3
  • E πœ‹ 3 βˆ’ √ 3

Q18:

Find the area of the region enclosed by one loop of the polar curve π‘Ÿ = 4 πœƒ s i n .

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

Q19:

Find the area inside the circle π‘Ÿ = 4 πœƒ c o s and outside the circle π‘Ÿ = 2 .

  • A 2 √ 3 + 4 3 πœ‹
  • B √ 3 βˆ’ 1 3 πœ‹
  • C √ 3 + 2 3 πœ‹
  • D 2 √ 3 βˆ’ 2 3 πœ‹
  • E 2 πœ‹

Q20:

Find the area of the region enclosed by one loop of the polar curve π‘Ÿ = 4 3 πœƒ c o s .

  • A 4 πœ‹ 3
  • B 2 πœ‹
  • C 8 πœ‹ 3
  • D 8 3
  • E 4 3

Q21:

Find the area of the region bounded by the polar curve π‘Ÿ = πœƒ + πœƒ s i n c o s , where 0 ≀ πœƒ ≀ πœ‹ .

  • A πœ‹ 2
  • B πœ‹
  • C 1 2
  • D2
  • E πœ‹ 4

Q22:

Find the area of the region common to the interior of π‘Ÿ = 4 ( 2 πœƒ ) s i n and π‘Ÿ = 2 .

  • A 4 3 ο€» 4 πœ‹ βˆ’ 3 √ 3 
  • B 2 3 ο€» 4 πœ‹ βˆ’ 3 √ 3 
  • C 4 ο€» 2 βˆ’ 3 √ 3 + πœ‹ 3 
  • D 4 ( 4 βˆ’ πœ‹ )
  • E 4 πœ‹

Q23:

Find the area of the region enclosed by one petal of π‘Ÿ = 4 ( 3 πœƒ ) c o s .

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

Q24:

Find the area enclosed by one loop of the rose with polar equation π‘Ÿ = 2 πœƒ c o s .

  • A πœ‹ 8
  • B1
  • C πœ‹ 1 6
  • D 1 4
  • E 1 2

Q25:

Find the area of the region enclosed by the polar curve π‘Ÿ = √ 1 + ( 5 πœƒ ) c o s  .

  • A 3 πœ‹ 2
  • B 3 πœ‹ 4
  • C 3 πœ‹
  • D πœ‹
  • E πœ‹ 2
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