Given information :
Horizontal line y=c is intersecting the curve y=8x−27x3 and the area R1=R2
Formula used :
∫abxndx=[xn+1n+1]ab
When y=c is intersecting the curve y=8x−27x3 , at intersecting point both the functional values are equal.
Calculation :
Given line is y=c ...........(i)
Given curve is y=8x−27x3 ..................(ii)
At the intersecting points of (i) and (ii) ,
8x−27x3=c ....................(iii)
The region R1 limes between x=0 and x=a and R2 limes between x=a and x=b
We have R1=R2 ................(i)
So, ∫0a[c−(8x−27x3)]dx=∫ab[(8x−27x3)−c]dx
⇒[cx−8x22+27x44]0a=[8x22−27x44−cx]ab
Now, substituting c by 8x−27x3 , we get,
[8x2−27x4−8x22+27x44]0a=[8x22−27x44−8x2+27x4]ab⇒[4x2−814x4]0a=[−4x2+814x4]0a⇒4a2−814a4=−4b2+4a2+814b4−814a4⇒b2(814b2−4)=0⇒814b2−4=0 [as b≠0]⇒b2=1681⇒b=49
When x=b , the equation (iii) be
c=f(b)=8b−27b3=8×49−27×649×9×9=329−6427=96−6427=3227