Examples: Find the area of the region bounded by the curve `y^2 = 2y - x` and the y-axis.

Explanation

`y^(2) = 2y - x implies y^(2) - 2y = -x `
`implies y^(2) - 2y + 1 = -x +1 `
`implies (y-1)^(2) = -(x-1)`
`implies Y^(2) = - X`

This is a left-handed parabola with vertex at (X = 0 , Y = 0) .
X = 0 , Y = 0 `rArr - x +1=0 and y-1 -0 `
`rArr x =1 and y=1 `
Thus , the vertex of the given parabola is A(1,1) Also `x=0 rarr y^2=0 rArr y(y-2)=0 rArr y=0 or y =2`
Thus , the curve meets the y-axis at O(0,0) and B(0,2)
a rough sketch of the curve can be drawn,as shown in the firgue .
`therefore " required area " = underset(0)overset(2)int"x dy " = underset(0)overset(2)int(2y-y^2)dy `
`=[y^2-y^2/3]_0^2 =(4-8/3)=4/3` sq units
Hence , the required area is `4/3` sq units