Find the area of the region bounded by the curve y2=2y-x and the y-axis.
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