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Question

The area in the positive quadrant is enclosed by the circle x2+y2=4, the line x=y3 and the x-axis is


A

π2

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B

π4

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C

π3

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D

π

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Solution

The correct option is C

π3


Explanation for correct answer:

Step 1:Representing the curve

Given the equation of a circle is

x2+y2=4(i)

The radius of this circle is 2 units.

Also.

x=y3y=x3(ii)

Put equation (ii) in equation (i), we get,

x2+x32=43x2+x23=44x2=12x2=124x2=3x=±3

Put x=3 in equation (ii) we get

y=33y=1

Therefore the point of intersection in the positive quadrant is 3,1

Step 2:Determining the area in the positive quadrant

The area of a shaded region is

A=03x3dx+324-x2dx=13x2203+12x4-x2+42sin-1x32=323-0+12×24-22+42sin-12-12×3×4-32-42sin-13=32+2×π2-32-2π3=π-2π3=π3

Therefore the area of the shadow region is equal to π3 square units.

Hence, the correct answer is option (C).


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