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Question
Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32 is ______.
Options
16π sq.units
4π sq.units
32π sq.units
24 sq.units
Solution
Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32 is 4π sq.units.
Explanation:
Given equation of circle is x2 + y2 = 32
⇒ x2 + y2 = `(4sqrt(2))^2` and the line is y = x and the x-axis.
Solving the two equations
We have x2 + x2 = 32
⇒ 2x2 = 32
⇒ x2 = 16
∴ x = ± 4
Required area = `int_0^4 x "d"x + int_4^(4sqrt(2)) sqrt((4sqrt(2))^2 - x^2) "d"x`
= `1/2 [x^2]_0^4 + [x/2 sqrt((4sqrt(2))^2 - x^2) + 32/2 sin^-1 x/(4sqrt(2))]_4^(4/sqrt(2))`
= `1/2 [16 - 0] + [0 + 16 sin^-1 ((4sqrt(2))/(4sqrt(2))) - 2sqrt(32 - 16) - 16sin^-1 4/(4sqrt(2))]`
= `8 + [16 sin^-1 (1) - 8 - 16sin^-1 1/sqrt(2)]`
= `8 + 16 * pi/2 - 8 - 16 * pi/4`
= `8pi - 4pi`
= 4π sq.units
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