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Worksheet: Area of Regions between Two Curves

Q1:

The figure shows 𝑦 = π‘₯ βˆ’ 6 π‘₯ + 1 1 π‘₯ βˆ’ 3 3 2 .

Evaluate the area of the shaded region giving your answer as a fraction.

  • A 3 4
  • B 1 3 4
  • C 1 1 4
  • D 1 4
  • E 2 7 4

Q2:

Find the area of the region bounded by 𝑦 = π‘₯ 3 and 𝑦 = π‘₯ .

  • A 1 6
  • B 3 2
  • C 5 6
  • D 1 2
  • E 4 3

Q3:

Find the area of the region bounded by the curves 𝑦 = 3 π‘₯ βˆ’ 5 π‘₯ 2 and 𝑦 = βˆ’ 5 π‘₯ 2 .

  • A 6 2 5 3 8 4
  • B 1 3 7 5 2 4
  • C 2 5 1 9 2
  • D 1 2 5 3 8 4
  • E 2 5 4 8

Q4:

Find the area of the region bounded above by 𝑦 = 2 π‘₯ and below by 𝑦 = 2 π‘₯ βˆ’ 5 π‘₯ 2 .

  • A 4 4 1 1 1 2
  • B 5 7 1 6
  • C 7 1 1 1 2 4
  • D 1 4 7 2 4
  • E 1 1 4 1 3

Q5:

Find the area of the region bounded by π‘₯ = 𝑦 4 , 𝑦 = βˆ’ √ βˆ’ 2 π‘₯ + 1 , and 𝑦 = 0 .

  • A βˆ’ 1 1 6 + 5 √ 2 2 4
  • B √ 2 8
  • C 1 1 6 + 5 √ 2 2 4
  • D 1 1 √ 2 6 0
  • E √ 2 3

Q6:

Find the area of the region enclosed by the curves 𝑦 = 5 π‘₯ and 𝑦 = ( 2 π‘₯ βˆ’ 5 ) 2 .

  • A 1 2 5 3
  • B 1 2 5 6
  • C 6 2 5 6
  • D 1 1 2 5 3 2
  • E 6 2 5 8

Q7:

The curves shown are 𝑦 = 1 π‘₯ and 𝑦 = 1 π‘₯ 2 . What is the area of the shaded region? Give an exact answer.

  • A 1 + ( 2 ) l n
  • B0.3068528194
  • C βˆ’ 0 . 3 0 6 8 5 2 8 1 9 4
  • D 1 βˆ’ ( 2 ) l n
  • E0.6931471806

Q8:

Find the area of the region bounded by π‘₯ = 𝑦 and 2 π‘₯ + 𝑦 = 3 2 .

  • A 6 4 3
  • B 2 8 3
  • C 2 9 6
  • D 1 6 3
  • E8

Q9:

Find the area of the region bounded by 𝑦 = 2 √ 2 π‘₯ 3 and 𝑦 = 1 4 π‘₯ 2 between π‘₯ = 0 and π‘₯ = 6 .

  • A4
  • B 3 2 3
  • C 5 2 3
  • D 2 0 3
  • E16

Q10:

Find the area of the region bounded by 𝑦 = √ π‘₯ βˆ’ 5 and π‘₯ βˆ’ 3 𝑦 = 3 .

  • A 9 1 6
  • B 5 5 6
  • C 1 5 1 6
  • D 1 6
  • E 1 3

Q11:

Find the area of the region bounded by π‘₯ = βˆ’ 5 𝑦 + 1 2 and π‘₯ = 2 𝑦 βˆ’ 5 2 .

  • A 1 2 √ 4 2 7
  • B 1 0 √ 3
  • C 8 0 √ 4 2 4 9
  • D 8 √ 4 2 7
  • E 5 9 √ 4 2 5 4

Q12:

The figure shows the area between 𝑦 = 1 π‘₯ and 𝑦 = 1 π‘₯ 2 and the area between π‘₯ = 3 4 and π‘₯ = 1 . 2 . It is apparent that the piece over interval [ 0 . 7 5 , 1 ] , which has area 𝐴 , is bigger than the piece over [ 1 , 1 . 2 ] of area 𝐡 1 . 2 .

Verify the claim by determining the function d ( π‘₯ ) = 𝐴 βˆ’ 𝐡 π‘₯ . What is d ( 1 . 2 ) to two decimal places?

  • A d l n d ( π‘₯ ) = 4 3 βˆ’ 1 π‘₯ βˆ’ ο€Ό 3 π‘₯ 4  , ( 1 . 2 ) = 0 . 6 1
  • B d l n d ( π‘₯ ) = 4 3 βˆ’ 1 π‘₯ βˆ’ ο€Ό 4 π‘₯ 3  , ( 1 . 2 ) = 0 . 1 7 2
  • C d l n d ( π‘₯ ) = 4 3 βˆ’ 1 π‘₯ βˆ’ ο€Ό 3 π‘₯ 4  , ( 1 . 2 ) = 0 . 7 4 2
  • D d l n d ( π‘₯ ) = 4 3 βˆ’ 1 π‘₯ βˆ’ ο€Ό 4 π‘₯ 3  , ( 1 . 2 ) = 0 . 0 3
  • E d l n d ( π‘₯ ) = 4 3 + 1 π‘₯ + ο€Ό 4 π‘₯ 3  , ( 1 . 2 ) = 0 . 0 3

What theorem will assure us that there is a 𝑧 ∈ [ 1 . 2 , 2 ] so that the area of the pieces to the left and right of π‘₯ = 1 have the same area?

  • Athe extreme value theorem since d ( 2 ) > 0
  • Bthe intermediate value theorem since d ( 2 ) > 0
  • Cthe extreme value theorem since d ( 2 ) < 0
  • Dthe intermediate value theorem since d ( 2 ) < 0

Using the Newton-Raphson method, determine 𝑧 to 5 decimal places after a single step starting at π‘₯ = 1 . 2 .

By repeated use of Newton-Raphson, find 𝑧 to 5 decimal places.

Q13:

Find the area of the region bounded by 𝑦 = π‘₯ 2 2 , 𝑦 = π‘₯ 2 , βˆ’ π‘₯ + 3 𝑦 = 4 , where π‘₯ β‰₯ 0 .

  • A 1 2 9 8 1
  • B 1 4 1 8 1
  • C 2 4 2 7
  • D 5 8 8 1
  • E 6 6 1 4 8 1

Q14:

Determine, to the nearest hundredth, the area of the region bounded by the curve 𝑦 = √ π‘₯ 1 , the line 𝑦 = π‘₯ βˆ’ 6 2 and the 𝑦 -axis.