Refer to the figure and find the volume generated by rotating the given region about the specified line. ℛ1 about AB

I guess that the area we care about is the yellow area, delimited by the functions.

f(x) = 8*(x)^(1/4)

and the line with the slope s= 8/1 = 8 (as the line goes through the points (0,0) and (1, 8)).

g(x) = 8*x

then we want tofind the area between x = 0 and x = 1, of f(x) - g(x)

then we have:

now, for the area under the g(x) we have:

then I - I2 = 6.4 - 4 = 2.4

The yellow area is 2.4

And then, if we rotate this about the line AB, the volume will be:

B = 2*pi*2.4 = 2*3.14*2.4 = 15.075

The figure will be something like a half spheroid, with a hole in the shape of a cone inside of it.