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Find the area bounded by the parabola x = 8 + 2y – y2; the y - axis and the lines y = – 1 and y = 3.
Given: - Two equation;
Parabola x = 8 + 2y – y2 ,
y - axis,
Line1 y = – 1, and Line2 y = 3
Now to find the area between these four curves, we have to find a common area (ABDC) or the shaded part.
The 1st intersection of a parabola with line y = – 1, we get,
Putting the value of y = 1 in parabolic equation
⇒ x = 8 + 2y – y2
⇒ x = 8 + 2( – 1) – 1
⇒ x = 5
Hence intersection point is D(5, – 1)
The 2nd intersection of parabola with y = 3
Putting the value of y in parabola equation
⇒ x = 8 + 2y – y2
⇒ x = 8 + 2(3) – 32
⇒ x = 8 + 6 – 9
⇒ x = 5
Hence, intersection point is C(5,3)
and other points are A(0,3), B(0, – 1)
From the figure, we can see that, By taking a horizontal strip
The area under shaded portion = Area under parabola from y =– 1 to y = 3.
Tip: - Take limits as per strips. If strip is horizontal than take y limits or if integrating concerning y then limits are of y.
Here, limits are for y i.e. from – 1 to 3
Now putting limits, we get,