Worksheet: Area between a Curve and a Line

In this worksheet, we will practice applying integration to find the area between the curve of a function and a horizontal or vertical straight line.

Q1:

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

  • A0.4054651081
  • B1.09861228866811
  • C1.584962501
  • D l n ( 3 )
  • E l n ( 2 )

Q2:

A glassed, arched entry to a hotel follows the curve 𝑦 = 1 2 ( 𝑥 9 ) ( 𝑥 3 ) , where 𝑦 is the vertical height of the arch at a horizontal distance of 𝑥 metres from a point on the floor. If glass costs 1‎ ‎190 LE per square metre, calculate the cost of the entrance.

Q3:

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

  • A1.09861228866811
  • B ( 3 ) l n
  • C 1 . 0 9 8 6 1 2 2 8 8 6 6 8 1 1
  • D l n ( 3 )
  • E l n ( 4 )

Q4:

The figure shows the graph of 𝑓 ( 𝑥 ) = 1 4 ( 𝑥 2 ) ( 𝑥 + 1 ) 2 .

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

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

Q5:

Determine the area of the plane region bounded by the curve 𝑦 = 8 𝑥 and the lines 𝑥 = 1 , 𝑥 = 8 , and 𝑦 = 0 rounded to the nearest hundredth.

Q6:

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

  • A1.09861228866811
  • B ( 3 ) l n
  • C 1 . 0 9 8 6 1 2 2 8 8 6 6 8 1 1
  • D l n ( 3 )
  • E ( 4 ) l n

Q7:

Let 𝑓 ( 𝑥 ) = 2 𝑥 + 3 . Determine the area bounded by the curve 𝑦 = 𝑓 ( 𝑥 ) , the 𝑥 -axis, and the two lines 𝑥 = 1 and 𝑥 = 5 .

  • A 9 0 square units
  • B 2 7 0 square units
  • C 2 8 4 3 square units
  • D 1 0 2 square units

Q8:

Determine the area of the plane region bounded by the curve 𝑦 = 3 𝑥 6 𝑥 + 9 and the 𝑥 -axis rounded to the nearest hundredth.

Q9:

The figure shows 𝑦 = 𝑥 6 𝑥 + 1 1 𝑥 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

Q10:

Calculate the area of the plane region bounded by the curve 𝑦 = 𝑥 + 6 𝑥 7 2 and the 𝑥 -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

Q11:

Find the area of the region above the 𝑥 -axis that is bounded by the graph of the function 𝑓 𝑓 ( 𝑥 ) = 8 𝑥 5 𝑥 + 3 and the line 𝑥 = 8 . Give your answer to two decimal places.

Q12:

Let 𝑓 𝑓 ( 𝑥 ) = 5 𝑥 + 1 5 . Determine the area bounded by the curve 𝑦 = 𝑓 ( 𝑥 ) , the 𝑥 -axis, and the line 𝑥 = 2 .

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.