Question

How do you find the volume of the solid generated by revolving the region enclosed by the parabola `y^2=4x` and the line y=x revolved about the y-axis?

Answer 1

`(128pi)/15`

Explanation:

Consider an horizontal strip of thickness `delta y` revolved around y axis. The volume of the solid so generated would be `pi x_1^2 delta y - pi x_2^2 deltay`. This is illustrated in the figure below.

The volume of the solid generated by revolving the who region bounded by the given parabola and the straight line would thus be

`int_(y=0) ^4 pi x_1^2 delta y - pi x_2^2 deltay`. Substituting `y= x_1` and `y^2 /4 = x_2`, te given integral would become

`pi int_0^4 (y^2 -y^4 /16 ) dy`

`pi [ y^3 /3 -y^5 /80]_0^4`

`pi[64/3 -1024/80]` = `64pi(1/3 -1/5)`= `(128pi)/15`