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Worksheet: Calculating the Area between Curves

Q1:

Calculate the area of the plane region bounded by the curve 𝑦 = π‘₯ + 6 π‘₯ βˆ’ 7 2 and the π‘₯ -axis.

  • A 2 2 3 area units
  • B 2 4 5 3 area units
  • C 1 1 3 area units
  • D 2 5 6 3 area units

Q2:

Find the area of the region enclosed by the curves 𝑦 = 𝑒 2 π‘₯ and 𝑦 = 2 π‘₯ βˆ’ 5 2 and the lines π‘₯ = βˆ’ 3 and π‘₯ = 1 .

  • A 𝑒 2 βˆ’ 1 2 βˆ’ 1 2 𝑒 2 6
  • B 𝑒 + 4 3 βˆ’ 1 𝑒 2 6
  • C 𝑒 βˆ’ 1 2 βˆ’ 1 𝑒 2 6
  • D 𝑒 2 + 4 3 βˆ’ 1 2 𝑒 2 6
  • E 𝑒 2 βˆ’ 8 3 βˆ’ 1 2 𝑒 2 6

Q3:

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

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

Q4:

The plan view of a single corridor floor is bounded by lines π‘₯ = 0 , 𝑦 = 0 and the curve 𝑦 = 5 π‘₯ 3 βˆ’ 1 5 2 , all measured in metres. What is the cost of covering 6 such corridors with granite at the price of 200 pounds per square metre?

Q5:

The plan view of a single corridor floor is bounded by lines π‘₯ = 0 , 𝑦 = 0 and the curve 𝑦 = 3 π‘₯ 4 βˆ’ 3 2 , all measured in metres. What is the cost of covering 3 such corridors with granite at the price of 100 pounds per square metre?

Q6:

Find the area of the region bounded above by 𝑦 = 1 π‘₯ , bounded below by 𝑦 = 1 2 π‘₯ 2 , and bounded on the side by π‘₯ = 1 .

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

Q7:

Find the area of the region bounded above by 𝑦 = 1 π‘₯ , bounded below by 𝑦 = 1 2 π‘₯ 2 , and bounded on the side by π‘₯ = 3 .

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

Q8:

Determine the area of the region bounded by the graphs of the functions 𝑓 and 𝑔 , where 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 1 2 , and 𝑔 ( π‘₯ ) = βˆ’ ( π‘₯ + 5 ) + 6 2 .

  • A 1 0 9 square units
  • B 5 7 square units
  • C 1 5 square units
  • D 9 square units
  • E 6 7 square units

Q9:

Determine the area of the region bounded by the graphs of the functions 𝑓 and 𝑔 , where 𝑓 ( π‘₯ ) = π‘₯ + 5 2 , and 𝑔 ( π‘₯ ) = βˆ’ ( π‘₯ + 6 ) + 2 5 2 .

  • A 1 1 2 square units
  • B 2 0 0 3 square units
  • C 6 0 square units
  • D 8 3 square units
  • E 1 2 4 square units

Q10:

Find the area of the region bounded by the curves 𝑦 = π‘₯ π‘₯ l n and 𝑦 = ( π‘₯ ) π‘₯ l n 2 .

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

Q11:

Find the area of the region bounded by the curves 𝑦 = 4 π‘₯ π‘₯ l n and 𝑦 = 3 ( π‘₯ ) π‘₯ l n 2 .

  • A8
  • B 1 6 0 2 7
  • C 8 0 3
  • D 3 2 2 7
  • E 6 4 2 7

Q12:

Determine, to the nearest thousandth, the area of the plane region bounded by the curve 𝑦 = √ 2 π‘₯ βˆ’ 2 and the lines π‘₯ = 2 , π‘₯ = 3 , and 𝑦 = 0 .

Q13:

Determine, to the nearest thousandth, the area of the plane region bounded by the curve 𝑦 = √ 7 π‘₯ βˆ’ 7 and the lines π‘₯ = 3 , π‘₯ = 4 , and 𝑦 = 0 .

Q14:

Calculate the area bounded by the graph of the function 𝑓 ( π‘₯ ) = ( 5 βˆ’ π‘₯ ) ( π‘₯ βˆ’ 1 ) 2 and the two coordinate axes.

  • A 1 9 1 2 square units
  • B 6 4 3 square units
  • C75 square units
  • D 2 7 5 1 2 square units
  • E 3 2 5 4 square units

Q15:

Calculate the area bounded by the graph of the function 𝑓 ( π‘₯ ) = ( 5 βˆ’ π‘₯ ) ( π‘₯ βˆ’ 3 ) 2 and the two coordinate axes.

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

Q16:

Find the area of the region bounded by 𝑦 = 2 βˆ’ | π‘₯ | and 𝑦 = π‘₯ 4 .

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

Q17:

Find the area of the region bounded by 𝑦 = 3 βˆ’ 2 | π‘₯ | and 𝑦 = π‘₯ 4 .

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

Q18:

Find the area of the region bounded by 𝑦 = 2 π‘₯ t a n and 𝑦 = 2 2 π‘₯ s i n between π‘₯ = βˆ’ πœ‹ 6 and π‘₯ = πœ‹ 6 .

  • A l n 2 + 1
  • B βˆ’ 1 2 2 + 1 l n
  • C 1 2 2 + 1 l n
  • D βˆ’ 2 + 1 l n
  • E l n 2 + 2

Q19:

Find the area of the region bounded by 𝑦 = 2 π‘₯ t a n and 𝑦 = 4 π‘₯ s i n between π‘₯ = βˆ’ πœ‹ 3 and π‘₯ = πœ‹ 3 .

  • A 4 2 + 4 l n
  • B βˆ’ 2 2 + 4 l n
  • C 2 2 + 4 l n
  • D βˆ’ 4 2 + 4 l n
  • E l n 2 + 4

Q20:

Determine the area of the plane region bounded by the curve 𝑦 = βˆ’ π‘₯ + 2 0 2 , the π‘₯ -axis, and the two lines π‘₯ = βˆ’ 3 and π‘₯ = 2 .

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

Q21:

Determine the area of the plane region bounded by the curve 𝑦 = βˆ’ π‘₯ + 2 2 2 , the π‘₯ -axis, and the two lines π‘₯ = βˆ’ 2 and π‘₯ = 4 .

  • A60 square units
  • B 7 6 3 square units
  • C16 square units
  • D108 square units