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.

  • A1(2)ln
  • B1+(2)ln
  • C0.3068528194
  • D0.6931471806
  • E0.3068528194

Q2:

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

  • A12
  • B56
  • C16
  • D32
  • E43

Q3:

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

  • A2548
  • B625384
  • C125384
  • D1,37524
  • E25192

Q4:

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

  • A5
  • B32
  • C16
  • D56
  • E13

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.

  • A413 square units
  • B65 square units
  • C2653 square units
  • D212 square units

Q8:

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

  • A556
  • B1516
  • C13
  • D916
  • E16

Q9:

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

  • A116+5224
  • B28
  • C116+5224
  • D23
  • E11260

Q10:

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

  • A296
  • B8
  • C163
  • D283
  • E643

Q11:

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

  • A594254
  • B12427
  • C8427
  • D804249
  • E103

Q12:

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

  • A1+2𝜋3+33
  • B4+4𝜋3
  • C2𝜋3+4
  • D3+3+2𝜋3
  • E2𝜋3+43

Q13:

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

  • A12
  • B324ln
  • C32+4ln
  • Dln4+2
  • Eln412

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.

  • A553320
  • B214
  • C2120
  • D257160
  • E25732

Q15:

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

  • A83+16𝜋3
  • B16𝜋3
  • C23+16𝜋3
  • D33+16𝜋3
  • E23+16𝜋3

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.

  • A1+𝜋2
  • B1+3𝜋8
  • C1+3𝜋4
  • D1+3𝜋8
  • E1+3𝜋4

Q18:

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

  • A5233
  • B3
  • C43
  • D123
  • E23

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.

  • A42ln
  • B48ln
  • C32+42ln
  • D42+72ln
  • E1+42ln

Q20:

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

  • A162ln
  • B22ln
  • C562ln
  • D122ln
  • E232ln

Q21:

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

  • A𝑒2+4312𝑒
  • B𝑒+431𝑒
  • C𝑒121𝑒
  • D𝑒21212𝑒
  • E𝑒28312𝑒

Q22:

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

  • A441112
  • B11413
  • C14724
  • D5716
  • E711124

Q23:

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

  • A1,12532
  • B1253
  • C6258
  • D1256
  • E6256

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.

  • A2+1ln
  • Bln2+6
  • Cln2+2
  • D12+2ln
  • E14+2ln

Q25:

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

  • A245
  • B335
  • C125
  • D1310
  • E235

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