Find the area of the region between the parabola x = 4y – y2 and the line x = 2y – 3.
Given: - Two equation; Parabola x = 4y – y2 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 – y2 =2y – 3
⇒ y2 – 2y – 3 = 0
⇒ y2 – 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