I have a few questions from this image:
As you can see, numbers 4 and 6 have the same equation. The only difference is that the axis of revolution are different. For 4, its x = 3, and for 6, its y = 6.
Will it be correct to use the same equation?
Because when I solved the problem, I used a different equation for number 6, since the axis of rotation is also different.
Here is what I used:
Shell Method: V = integral 2*pi*r*th from a to b
r = x
t = dx
h = y(curve) - y(line) = 4x-x^-x = 3x-x^2
a=0, b = 3
therefore V= 2*pi integral x(3x-x^2)dx from 0 to 3, which gives 27pi/2.
Now, for the Centroid, Mx = Vy(bar) = integral (2*pi*3x^2-x^3)y(bar) dx from 0 to 3
y(bar) = (ycurve + yline) /2 = 5x - x^2
Mx = integral pi*(3x^2 - x^3)(5x - x^2) dx from 0 to 3, which gives an answer of 729pi/20, different from the 567pi/20 from the image. What am I doing wrong?