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In this lesson, we will learn how to calculate the areas of enclosed regions.

Q1:

The figure shows π¦ = π₯ β 6 π₯ + 1 1 π₯ β 3 3 2 .

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

Q2:

Find the area of the region bounded by π¦ = π₯ 3 and π¦ = π₯ .

Q3:

Find the area of the region bounded by the curves π¦ = 3 π₯ β 5 π₯ 2 and π¦ = β 5 π₯ 2 .

Q4:

Find the area of the region bounded above by π¦ = 2 π₯ and below by π¦ = 2 π₯ β 5 π₯ 2 .

Q5:

Find the area of the region bounded by π₯ = π¦ 4 , π¦ = β β β 2 π₯ + 1 , and π¦ = 0 .

Q6:

Find the area of the region enclosed by the curves π¦ = 5 π₯ and π¦ = ( 2 π₯ β 5 ) 2 .

Q7:

The curves shown are π¦ = 1 π₯ and π¦ = 1 π₯ 2 . What is the area of the shaded region? Give an exact answer.

Q8:

Find the area of the region bounded by π₯ = π¦ and 2 π₯ + π¦ = 3 2 .

Q9:

Find the area of the region bounded by π¦ = 2 β 2 π₯ 3 and π¦ = 1 4 π₯ 2 between π₯ = 0 and π₯ = 6 .

Q10:

Find the area of the region bounded by π¦ = β π₯ β 5 and π₯ β 3 π¦ = 3 .

Q11:

Find the area of the region bounded by π₯ = β 5 π¦ + 1 2 and π₯ = 2 π¦ β 5 2 .

Q12:

The figure shows the area between π¦ = 1 π₯ and π¦ = 1 π₯ 2 and the area between π₯ = 3 4 and π₯ = 1 . 2 . It is apparent that the piece over interval [ 0 . 7 5 , 1 ] , which has area π΄ , is bigger than the piece over [ 1 , 1 . 2 ] of area π΅ 1 . 2 .

Verify the claim by determining the function d ( π₯ ) = π΄ β π΅ π₯ . What is d ( 1 . 2 ) to two decimal places?

What theorem will assure us that there is a π§ β [ 1 . 2 , 2 ] so that the area of the pieces to the left and right of π₯ = 1 have the same area?

Using the Newton-Raphson method, determine π§ to 5 decimal places after a single step starting at π₯ = 1 . 2 .

By repeated use of Newton-Raphson, find π§ to 5 decimal places.

Q13:

Find the area of the region bounded by π¦ = π₯ 2 2 , π¦ = π₯ 2 , β π₯ + 3 π¦ = 4 , where π₯ β₯ 0 .

Q14:

Determine, to the nearest hundredth, the area of the region bounded by the curve π¦ = β π₯ 1 , the line π¦ = π₯ β 6 2 and the π¦ -axis.

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