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=1𝑥 and 𝑦=1𝑥. What is the area of the shaded region? Give an exact answer.

  • A 1 ( 2 ) l n
  • B 1 + ( 2 ) l n
  • C0.3068528194
  • D0.6931471806
  • E 0 . 3 0 6 8 5 2 8 1 9 4

Q2:

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

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

Q3:

Find the area of the region bounded by the curves 𝑦=3𝑥5𝑥 and 𝑦=5𝑥.

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

Q4:

Find the area of the region bounded by the curves 𝑦=𝑥𝑥ln and 𝑦=(𝑥)𝑥ln.

  • A5
  • B 3 2
  • C 1 6
  • D 5 6
  • E 1 3

Q5:

Determine, to the nearest thousandth, the area of the plane region bounded by the curve 𝑦=2𝑥2 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=5𝑥315, all measured in meters. What is the cost of covering 6 such corridors with granite at the price of 200 pounds per square meters?

Q7:

Determine the area of the plane region bounded by the curve 𝑦=𝑥+20, the 𝑥-axis, and the two lines 𝑥=3 and 𝑥=2.

  • A 4 1 3 square units
  • B65 square units
  • C 2 6 5 3 square units
  • D 2 1 2 square units

Q8:

Find the area of the region bounded by 𝑦=𝑥5 and 𝑥3𝑦=3.

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

Q9:

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

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

Q10:

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

  • A 2 9 6
  • B8
  • C 1 6 3
  • D 2 8 3
  • E 6 4 3

Q11:

Find the area of the region bounded by 𝑥=5𝑦+1 and 𝑥=2𝑦5.

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

Q12:

Find the area of the region bounded by 𝑦=𝑥cos and 𝑦=3𝑥+2cos, where 0𝑥𝜋.

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

Q13:

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

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

Q14:

The curve in the figure is 𝑦=15𝑥3𝑥+4.

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

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

Q15:

Find the area of the region bounded by 𝑦=3𝑥4cos and 𝑦=5𝑥cos, where 0𝑥2𝜋.

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

Q16:

Determine, to the nearest thousandth, the area of the region bounded by the graph of the function 𝑓𝑓(𝑥)=(𝑥8)(𝑥3)(𝑥2), 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=𝑥sin, 𝑥=𝜋2, and 𝑥=𝜋.

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

Q18:

Find the area of the region enclosed by the curves 𝑦=16𝑥cos and 𝑦=2𝑥sec for 𝑥 between 𝜋3 and 𝜋3.

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

Q19:

Consider the region in the first quadrant enclosed by the curves 𝑦=4𝑥, 𝑦=𝑥, and 𝑦=𝑥4. Find the area of this region.

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

Q20:

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

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

Q21:

Find the area of the region enclosed by the curves 𝑦=𝑒 and 𝑦=2𝑥5 and the lines 𝑥=3 and 𝑥=1.

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

Q22:

Find the area of the region bounded above by 𝑦=2𝑥 and below by 𝑦=2𝑥5𝑥.

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

Q23:

Find the area of the region enclosed by the curves 𝑦=5𝑥 and 𝑦=(2𝑥5).

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

Q24:

Find the area of the region bounded above by 𝑦=1𝑥, bounded below by 𝑦=12𝑥, and bounded on the side by 𝑥=1.

  • A 2 + 1 l n
  • B l n 2 + 6
  • C l n 2 + 2
  • 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 2 4 5
  • B 3 3 5
  • C 1 2 5
  • D 1 3 1 0
  • E 2 3 5

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