Find the area of the region bounded by the curve x = 25-y2, the Y-axis lying in the first quadrant and the lines y = 0 and y = 5 - Mathematics and Statistics

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Sum

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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Chapter 1.7: Application of Definite Integration - Q.2

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