Question

Find the area of the region bounded by the ellipse `(x^2)/4+(y^2)/9=1`

Answer 1

Given curve is an ellipse which is symmetric about X-axis and both, as shown in the figure.

` :. ` Area bounded by ellipse
`=4xx` (area of shaded region in first quadrant)
`=4 xx int_(0)^(2)ydx=4 int_(0)^(2)(3)/(2)sqrt(4-x^(2))dx`
`=6int_(0)^(2)sqrt(2^(2)-x^(2))dx`
`=6[(x)/(2)sqrt(4-x^(2))+(2^(2))/(2)"sin"^(-1)((x)/(2))]_(0)^(2)`
`=6{0+2"sin"^(-1)(1)-0}`
`=6xx2xx((pi)/(2))`
`=6pi` sq. units.