Answer :

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








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