# Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y^2=x and x=2y about the y-axis. Volume =?im having trouble solving this can someone...

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y^2=x and x=2y about the y-axis.

Volume =?

im having trouble solving this can someone show me how to do this

jeew-m | Certified Educator

The two graphs can be shown as in the figure.

y=x/2

y=sqrt(x)

The intersections can be obtain;

`sqrtx = x/2`

`x = x^2/4`

`0 = x^2-4x`

`0 = x(x-4)`

So the interceptions are at x=0 and x= 4

Coresponding y values are y=0 and y = 2

When we rotate the bound area around y, we will get a solid which is like a cone. But it has a internal diameter and external diameter.

Internal diameter is obtained by the y^2 = x graph and outer diameter is obtained by y=x/2 graph.  And y axis is symatrical.

Since y-axis is symetrical;

Internal radius of solid = length from y-axis to y^2 = x graph

= y^2

outer radius of solid    = length from y-axis to y=x/2 grapgh

= 2y

So the area of a small portion of the solid = `pi[(2y)^2-((y^2)^2)]`

Since the integration will be around y-axis the interval will be y cordinates of the intersepting points.

Volume = `int A(y)dy`

= `int_0^2 pi[(2y)^2-((y^2)^2)] dy`

= `pi*[4y^3/3-y^5/5]_0^2`

= 13.404

So the volume of solid encountered by the bound area of `y^2=x` and `2y=x` graphs is 13.404.