Find the area of the region bounded by the parabola y2 = 2x and the straight-line x – y = 4.


Given: -

Two equation;


Parabola y2 = 2x and


Line x – y = 4



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 line curve – Area under parabola; horizontal strip


Now, Intersection points;


From parabola and line equation equate y, x – 4 = y we get


y2 = 2x


(x – 4)2 = 2x


x2 – 8x + 16 = 2x


x2 – 10x + 16 = 0


x2 – 8x – 2x + 16 = 0


x(x – 8) – 2(x – 8) = 0


(x – 8)(x – 2) = 0


x = 8,2


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


y = x – 4


For x = 8


y = 8 – 4


y = 4


For x = 2


y = 2 – 4


y = – 2


Therefore, two intersection points coordinates are (8, 4) and (2, – 2)


Area of the bounded region


= Area under the line curve from – 2 to 4 – Area under parabola from – 2 to 4


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


Area bounded by region = {Area under line from – 2 to 4} – {Area under parabola from – 2 to 4}






Putting limits, we get





= 6 + 24 – 12


= 18 sq units


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