Question

How do i find the volume of this solid?

if it revolves around the `x` axis we integrate with respect to `x`. and the same goes for `y`. but what do i do when it revolves around `y=x`

Answer 1

by using the rotation of axes

Explanation:

the slope of the line `y=x` is 1 as it makes and angle of `pi/4rad`
with the `x`-axis so by using the formula

`x=u*cos(pi/4)-v*sin(pi/4)`

`y=u*sin(pi/4)+v*cos(pi/4)`

Where u,v are the new axes
so if `y=x^2`
so by substituting
`u*sin(pi/4)+v*cos(pi/4)`=`(u*cos(pi/4)-v*sin(pi/4))^2`

and You will get the equation of the parabola with respect to the new axes

You also use the equations above to find u,v co_ordinates of the points from which you will revolve the curve around on the new axis which will be `(0,0)` and `(sqrt(2),0)`
and by applying the integration formula:

I took this graph from http://www.wolframalpha.com/widgets/

I hope this was helpful.