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Lesson Worksheet: Area between a Curve and a Line Mathematics • Higher Education

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:

Let 𝑓(π‘₯)=2π‘₯+3. Determine the area bounded by the curve 𝑦=𝑓(π‘₯), the π‘₯-axis, and the two lines π‘₯=βˆ’1 and π‘₯=5.

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

Q2:

The figure shows 𝑦=π‘₯βˆ’6π‘₯+11π‘₯βˆ’3.

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

  • A114
  • B134
  • C14
  • D274
  • E34

Q3:

Determine the area of the plane region bounded by the curve 𝑦=βˆ’π‘₯+20, the π‘₯-axis, and the two lines π‘₯=βˆ’3 and π‘₯=2.

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

Q4:

Find the area enclosed by the graph of π‘₯=9βˆ’π‘¦οŠ¨, the 𝑦-axis, and the lines 𝑦=βˆ’3 and 𝑦=3.

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

Q5:

The curve shown is 𝑦=1π‘₯. What is the area of the shaded region? Give an exact answer.

  • A1.09861228866811
  • Bln(3)
  • Cβˆ’1.09861228866811
  • Dβˆ’(3)ln
  • Eβˆ’(4)ln

Q6:

Find the area of the shaded region.

Q7:

The figure shows the graph of the function 𝑓(π‘₯)=2π‘₯βˆ’8π‘₯. Evaluate the area of the shaded region.

Q8:

Find the area of the region above the π‘₯-axis bounded by the curve 𝑦=3π‘’ο‘οŽ¦ and the lines π‘₯=0 and π‘₯=14. Give an exact answer.

  • A21ο€Ήπ‘’βˆ’1ο…οŠ¨
  • B21π‘’οŠ¨
  • C7π‘’οŠ¨
  • D3ο€Ήπ‘’βˆ’1ο…οŠ¨
  • E3ο€Ή1βˆ’π‘’ο…οŠ¨

Q9:

The curve in the figure is 𝑦=15ο€Ήπ‘₯βˆ’3π‘₯+4ο…οŠ©οŠ¨.

What is the area of the shaded region? Give your answer exactly as a fraction.

  • A553320
  • B214
  • C2120
  • D257160
  • E25732

Q10:

Let π‘“βˆΆπ‘“(π‘₯)=√5π‘₯+15. Determine, to the nearest thousandth, the area bounded by the curve 𝑦=𝑓(π‘₯), the π‘₯-axis, and the line π‘₯=βˆ’2.

This lesson includes 51 additional questions and 167 additional question variations for subscribers.

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