The value of c
Equation of the parabola is y=8x-27x3
Let x1,c and x2,c be the intersection points of y=c and y=8x-27x3.
Let the area of the region below line y=c be R1 and the area of the region above line y=c be R2.
We need to find value of c where area of region R1 and area of region R2 is the same.
So assume that both areas are the same.
Setting both area in terms of integration, it becomes
∫0x1c-8x-27x3 dx= ∫x1x28x-27x3-cdx
Substituting the limits,
cx1-8x122+27x144=8x222-27x244-cx2-8x122+27x144+ c x1
Cancelling out common terms from both sides,
Now, we need to find intersection points of y=c and y=8x-27x3. At intersection
Since the point x2,c is one of the intersection points, substitute it in c=8x-27x3
Substituting this in the above simplified equation,
Factoring out x22,
Factoring it out,
x2=0 or x2=±49 From the figure, it can be seen that x2 cannot be 0 or negative.
The value of c is 3227