Worksheet: Area between Curves

In this worksheet, we will practice applying integration to find the area bounded by the curves of two or more functions.

Q1:

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

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

Q2:

Find the area of the region bounded by 𝑦 = 𝑥 and 𝑦 = 𝑥 .

  • 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 𝑥 5 𝑥 and 𝑦 = 5 𝑥 .

  • 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 by the curves 𝑦 = 𝑥 𝑥 l n and 𝑦 = ( 𝑥 ) 𝑥 l n .

  • A 3 2
  • B 5 6
  • C5
  • D 1 6
  • E 1 3

Q5:

Determine, to the nearest thousandth, the area of the plane region bounded by the curve 𝑦 = 2 𝑥 2 and the lines 𝑥 = 2 , 𝑥 = 3 , and 𝑦 = 0 .

Q6:

The plan view of a single corridor floor is bounded by lines 𝑥 = 0 , 𝑦 = 0 and the curve 𝑦 = 5 𝑥 3 1 5 , all measured in metres. What is the cost of covering 6 such corridors with granite at the price of 200 pounds per square metres?

Q7:

Determine the area of the plane region bounded by the curve 𝑦 = 𝑥 + 2 0 2 , the 𝑥 -axis, and the two lines 𝑥 = 3 and 𝑥 = 2 .

  • A65 square units
  • B 4 1 3 square units
  • C 2 1 2 square units
  • D 2 6 5 3 square units

Q8:

Find the area of the region bounded by 𝑦 = 𝑥 5 and 𝑥 3 𝑦 = 3 .

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

Q9:

Find the area of the region bounded by 𝑥 = 𝑦 , 𝑦 = 2 𝑥 + 1 , and 𝑦 = 0 .

  • A 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

Q10:

Find the area of the region bounded by 𝑥 = 𝑦 and 2 𝑥 + 𝑦 = 3 .

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

Q11:

Find the area of the region bounded by 𝑥 = 5 𝑦 + 1 and 𝑥 = 2 𝑦 5 .

  • 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:

Find the area of the region bounded by 𝑦 = 𝑥 c o s and 𝑦 = 3 𝑥 + 2 c o s , where 0 𝑥 𝜋 .

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

Q13:

The curves shown are and . What is the area of the shaded region? Give an exact answer.

  • A
  • B
  • C
  • D
  • E

Q14:

The curve in the figure is 𝑦 = 1 5 𝑥 3 𝑥 + 4 .

What is the area of the shaded region? Give your answer exactly as a fraction.

  • A 2 5 7 3 2
  • B 2 1 2 0
  • C 2 1 4
  • D 2 5 7 1 6 0
  • E 5 5 3 3 2 0

Q15:

Find the area of the region bounded by 𝑦 = 3 𝑥 4 c o s and 𝑦 = 5 𝑥 c o s , where 0 𝑥 2 𝜋 .

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

Q16:

Determine, to the nearest thousandth, the area of the region bounded by the graph of the function 𝑓 𝑓 ( 𝑥 ) = ( 𝑥 8 ) ( 𝑥 3 ) ( 𝑥 2 ) , where 𝑓 ( 𝑥 ) 0 , and the lines 𝑥 = 9 and 𝑦 = 0 .

Q17:

Find the area of the region enclosed by the curves 𝑦 = 𝑥 , 𝑦 = 𝑥 s i n , 𝑥 = 𝜋 2 , and 𝑥 = 𝜋 .

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

Q18:

Find the area of the region enclosed by the curves 𝑦 = 1 6 𝑥 c o s and 𝑦 = 2 𝑥 s e c for 𝑥 between 𝜋 3 and 𝜋 3 .

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

Q19:

Consider the region in the first quadrant enclosed by the curves 𝑦 = 4 𝑥 , 𝑦 = 𝑥 , and 𝑦 = 𝑥 4 . Find the area of this region.

  • A 3 2 + 4 l n 2
  • B 1 + 4 l n 2
  • C 4 l n 2 + 7 2
  • D 4 l n 2
  • E 4 l n 8

Q20:

Find the area of the region bounded by the curves 𝑦 = 𝑥 𝑥 + 1 and 𝑦 = 𝑥 𝑥 + 1 .

  • A 2 3 l n 2
  • B 5 6 l n 2
  • C 1 2 l n 2
  • D 1 6 l n 2
  • E 2 l n 2

Q21:

Find the area of the region enclosed by the curves 𝑦 = 𝑒 and 𝑦 = 2 𝑥 5 and the lines 𝑥 = 3 and 𝑥 = 1 .

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

Q22:

Find the area of the region bounded above by 𝑦 = 2 𝑥 and below by 𝑦 = 2 𝑥 5 𝑥 .

  • 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

Q23:

Find the area of the region enclosed by the curves 𝑦 = 5 𝑥 and 𝑦 = ( 2 𝑥 5 ) .

  • 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

Q24:

Find the area of the region bounded above by 𝑦 = 1 𝑥 , bounded below by 𝑦 = 1 2 𝑥 , and bounded on the side by 𝑥 = 1 .

  • A l n 2 + 6
  • B l n 2 + 2
  • C 2 + 1 l n
  • D 1 2 + 2 l n
  • E 1 4 + 2 l n

Q25:

Find the area of the region bounded by 𝑦 = 2 | 𝑥 | and 𝑦 = 𝑥 .

  • A 1 3 1 0
  • B 2 4 5
  • C 1 2 5
  • D 2 3 5
  • E 3 3 5

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