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Find the area bounded by the ellipse (x^...

Find the area bounded by the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` and the ordinates `x=a e\ a n d\ x=0,` where `b^2=a^2(1-e^2)a n d\ e<1.`

Text Solution

Verified by Experts

Required area A is given by
`=>A=2` (Area of shaded region in first quadrant)
`⇒A=2∫_0^(ae) |y|dx`
`⇒A=2(b)/a ∫_0^(ae)√(a^2-x^2) dx`
`⇒A=(2b)/a​[1/2​×sqrt(a^2−x^2) ​+1/2​a^2sin^(−1)(x/a)​]^(ae)​`
`⇒A=(2b)/a​[(ae)/2 sqrt(a^2−a^2e^2) ...
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