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Find the area of the region bounded by the curve y = `sqrt(9 - x^2)`, X-axis and lines x = 0 and x = 3

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#### Solution

Let A be the required area.

Given equation of the curve is y = `sqrt(9 - x^2)`

∴ A = `int_0^3 y "d"x`

= `int_0^3 sqrt(9 - x^2) "d"x`

= `int_0^3 sqrt((3)^2 - x^2) "d"x`

= `[x/2 sqrt((3)^2 - x^2) + (3)^2/2 sin^-1 (x/3)]_0^3`

= `[3/2 sqrt((3)^2 - (3)^2) + (3)^2/2 sin^-1 (3/3)] - [0/2 sqrt((3)^2 - 0^2) + (3)^2/2 sin^-1 (0/3)]`

= `0 + 9/2 sin^-1 (1) - 0`

= `9/2 (pi/2)`

∴ A = `(9pi)/4` sq.units

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