Question

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

Answer 1

Answer:

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:

$I = \int\limits^1_0 {f(x)} \, dx = \int\limits^1_0 {8*\sqrt[4]{x} )} \, dx = (8*(4/5)*\sqrt[4]{1^5} - 8*(4/5)*\sqrt[4]{0^5}) = 6.4$

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

$I2 = \int\limits^1_0 {g(x)} \, dx = \int\limits^1_0 {8x} \, dx = (8/2)*1^2 - (8/2)*0^2 = 4.$

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.