Find the area bounded by the parabola x = 8 + 2y – y^{2}; the y - axis and the lines y = – 1 and y = 3.

Given: - Two equation;

Parabola x = 8 + 2y – y^{2} ,

y - axis,

Line1 y = – 1, and Line2 y = 3

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Now to find the area between these four curves, we have to find a common area (ABDC) or the shaded part.

The 1^{st} intersection of a parabola with line y = – 1, we get,

Putting the value of y = 1 in parabolic equation

⇒ x = 8 + 2y – y^{2}

⇒ x = 8 + 2( – 1) – 1

⇒ x = 5

Hence intersection point is **D(5, – 1)**

The 2^{nd} intersection of parabola with y = 3

Putting the value of y in parabola equation

⇒ x = 8 + 2y – y^{2}

⇒ x = 8 + 2(3) – 3^{2}

⇒ 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,

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