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Question
Find the area of the region bounded by the curve x = `sqrt(25 - y^2)`, the Y-axis lying in the first quadrant and the lines y = 0 and y = 5
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Solution
Let A be the required area.
Given equation of the curve is x = `sqrt(25 - y^2)`
∴ A = `int_0^5 x "d"y`
= `int_0^5 sqrt(25 - y^2) "d"y`
= `int_0^5 sqrt((5)^2 - y^2) "d"y`
= `[y/2 sqrt((5)^2 - y^2) + (5)^2/2 sin^-1 (y/5)]_0^5`
= `[5/2 sqrt((5)^2 - (5)^2) + (5)^2/2 sin^-1 (5/5)] - [0/2 sqrt((5)^2 - 0) + (5)^2/2 sin^-1 (0/5)]`
= `0 + 25/2 sin^-1 (1) - 0`
= `25/2 (pi/2)`
= `(25pi)/4` sq.units
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