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Find the area enclosed by the ellipse (...

Find the area enclosed by the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`.

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given equation is
`(x^2)/(a^2)+(y^2)/(b^2)=1`.
`y^2 = b^2/(a^2) (a^2 - x^2)`
`y =b/a ( a^2 - x^2)^(1/2)`,
Now in the `i^(st)` quadrant
we know A= `4int _0^a ydx`
= `4int _0^a b/a sqrt(a^2-x^2)dx`
= `(4b)/aint _0^a sqrt(a^2-x^2)dx`
= `(4b)/a[ x/(2) sqrt(a^2-x^2) + a^2/(2) sin^-1(x/a)]_0^a`
= `(4b)/a[ a^2/(2) sin^-1 -0]`
= `(4b)/a. a^2/2 . pi/2`
= `piab` sq. units
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