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Question
Sketch the region `{(x, 0) : y = sqrt(4 - x^2)}` and x-axis. Find the area of the region using integration.
Solution
Given that `{(x, 0) : y = sqrt(4 - x^2)}`
⇒ y2 = 4 – x2
⇒ x2 + y2 = 4 which is a circle.
Required area = `2 * int_0^2 sqrt(4 - x^2) "d"x`
Since circle is symmetrical about y-axis
= `2 * int_0^2 sqrt((2)^2 - x^2) "d"x`
= `2 * [x/2 sqrt(4 - x^2) + 4/2 sin^-1 x/2]_0^2`
= `2[(2/2 sqrt(4 - 4) + 2 sin^-1 (1)) - (0 + 0)]`
= `2[2 * pi/2]`
= 2π sq.units
Hence, the required area = 2π sq.units
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