Lesson: Area between Curves and the x-Axis

In this lesson, we will learn how to calculate the areas of enclosed regions between a given curve and the x-axis as well as between specified bounds.

Worksheet: Area between Curves and the x-Axis • 8 Questions

Q1:

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

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.

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.

Q5:

Determine the area of the plane region bounded by the curve 𝑦 = 8 𝑥 4 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.

Q7:

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

Q8:

Determine the area of the plane region bounded by the curve 𝑦 = − 3 𝑥 − 6 𝑥 + 9 2 and the 𝑥 -axis rounded to the nearest hundredth.

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