Answer :

Given the boundaries of the area O befound are,


• Curve is y2 = 2y – x


• Y-axis.


Consider the curve, y2 = 2y –x


y2 – 2y = -x


by adding 1 on both sides


y2 – 2y + 1 = -(x-1)


(y-1)2 = -(x-1)


From the above equation, we can say that, the given equation is that of a parabola with vertex at A(1,1)


Consider the line x = 0 which is the y-axis, now substituting x = 0 in the curve equation we get


y2- 2y = 0


y(y-2)=0


y = 0 (or)y = 2


So , the parabola meets the y-axis at 2 points, B (0,2) and •(0,0)



As per the given boundaries,


• The parabola y2 = 2y-x, with vertex at A(1,1).


• X= 0 which is the y-axis.


The boundaries of the region to be found are,


Point A, where the curve y2 = 2y-x has the extreme end the vertex i.e. A (1,1)


Point O, which is the origin


Point B, where the curve y2 = 2y-x and the y – axis meet i.e. B (0,2)


Consider the curve,


y2 = 2y - x


x = 2y - y2


Area of the required region = Area of OBAO.





[Using the formula ]



The Area of the required region


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