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.

  • Aln(2)
  • Bln(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
  • B1.09861228866811
  • C(3)ln
  • Dln(4)
  • Eln(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.

  • A274
  • B54
  • C2716
  • D1316
  • E516

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
  • Bln(3)
  • C1.09861228866811
  • D(3)ln
  • E(4)ln

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.

  • A102square units
  • B270square units
  • C90square units
  • D2843square 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.

  • A114
  • B134
  • C14
  • D274
  • E34

Q10:

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

  • A2453 area units
  • B223 area units
  • C2563 area units
  • D113 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, to the nearest thousandth, 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.

  • A21𝑒1
  • B21𝑒
  • C7𝑒
  • D3𝑒1
  • E31𝑒

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.

Q16:

Find the area enclosed by the graph of 𝑥=9𝑦, the 𝑦-axis, and the lines 𝑦=3 and 𝑦=3.

  • A18 square units
  • B72 square units
  • C36 square units
  • D0 square units
  • E9 square units

Q17:

Find the area of the shaded region.

Q18:

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

Q19:

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.

Q20:

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?

Q21:

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.

Q22:

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

Q23:

Calculate the area bounded by the graph of the function 𝑓(𝑥)=(5𝑥)(𝑥1) and the two coordinate axes.

  • A643 square units
  • B27512 square units
  • C3254 square units
  • D75 square units
  • E1912 square units

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