Find the area of the region between the parabola x = 4y – y^{2} and the line x = 2y – 3.

Given: - Two equation; Parabola x = 4y – y^{2} and Line x = 2y – 3

Now to find an area between these two curves, we have to find a common area or the shaded part.

From figure, we can see that,

Area of shaded portion = Area under the parabolic curve – Area under line

Now, Intersection points;

From parabola and line equation equate x, we get

⇒ 4y – y^{2} =2y – 3

⇒ y^{2} – 2y – 3 = 0

⇒ y^{2} – 3y + y – 3 = 0

⇒ y(y – 3) + 1(y – 3)

⇒ (y + 1)(y – 3)

⇒ y = – 1,3

So, by putting the value of x in any curve equation, we get,

⇒ x = 2y – 3

For y = – 1

⇒ x = 2( – 1) – 3

⇒ x = – 5

For

y = 3

⇒ x = 2(3) – 3

⇒ x = 3

Therefore two intersection points coordinates are ( – 5, – 1) and (3, 3)

Area of the bounded region

= (Area under the parabola curve from – 1 to 3) – (Area under line from – 1 to 3)

**Tip: -** **Take limits as per strips. If strip is horizontal than take y limits or if integrating with respect to y then limits are of y.**

Here, limits are for y i.e from - 1 to 3.

Now putting limits, we get

1