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

  • A l n ( 2 )
  • B l n ( 3 )
  • C1.584962501
  • D0.4054651081
  • E1.09861228866811

Q2:

A glassed, arched entry to a hotel follows the curve 𝑦=12(𝑥9)(𝑥3), where 𝑦 is the vertical height of the arch at a horizontal distance of 𝑥 meters from a point on the floor. If glass costs 1,190 LE per square meter, calculate the cost of the entrance.

Q3:

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

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

Q4:

The figure shows the graph of 𝑓(𝑥)=14(𝑥2)(𝑥+1).

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

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

Q5:

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

Q6:

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

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

Q7:

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

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

Q8:

Determine the area of the plane region bounded by the curve 𝑦=3𝑥6𝑥+9 and the 𝑥-axis rounded to the nearest hundredth.

Q9:

The figure shows 𝑦=𝑥6𝑥+11𝑥3.

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

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

Q10:

Calculate the area of the plane region bounded by the curve 𝑦=𝑥+6𝑥7 and the 𝑥-axis.

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

Q11:

Find the area of the region above the 𝑥-axis that is bounded by the graph of the function 𝑓𝑓(𝑥)=8𝑥5𝑥+3 and the line 𝑥=8. Give your answer to two decimal places.

Q12:

Let 𝑓𝑓(𝑥)=5𝑥+15. Determine the area bounded by the curve 𝑦=𝑓(𝑥), the 𝑥-axis, and the line 𝑥=2.

Q13:

Find the area of the region above the 𝑥-axis bounded by the curve 𝑦=3𝑒 and the lines 𝑥=0 and 𝑥=14. Give an exact answer.

  • A 2 1 𝑒 1
  • B 2 1 𝑒
  • C 7 𝑒
  • D 3 𝑒 1
  • E 3 1 𝑒

Q14:

The figure shows the curve 𝑓(𝑥)=(𝑥2)19(𝑥2)+3, which has a horizontal tangent at the point 𝑃. Find the shaded area bounded by the curve 𝑓(𝑥) and the horizontal tangent. Give your answer to 1 decimal place.

Q15:

The figure shows the graph of the function 𝑓(𝑥)=2𝑥8𝑥. Evaluate the area of the shaded region.

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