From a point (x0, x02) on y= x2, find the line through that point perpendicular to y= x: y= -(x- x0)+ x02. Find where that line crosses y= x and calculate the distance from (x0, x02) to the point of intersection, as a function of x0. That will be the radius of the disk. Since the length of the segment y= x for y from x0 to x0+ \Delta x is \sqrt{2}\Delta x , in place of dx you will want to use \sqrt{2}dx as the "thickness" of the disk when you integrate.
Alternatively, you can rotate the graph through -45 degrees to make the line y= x rotate to y= 0. Since, in this case, x= x' cos 45- y'sin 45, y= x' sin 45+ y' cos 45 or x= \frac{\sqrt{2}}{2}x'+ \frac{\sqrt{2}}{2}y', y= \frac{\sqrt{2}}{2}x'- \frac{\sqrt{2}}{2}y'. Replacing x and y in y= x with those gives y= \frac{\sqrt{2}}{2}x'- \frac{\sqrt{2}}{2}y'= \frac{\sqrt{2}}{2}x'+ \frac{\sqrt{2}}{2}y' which reduces to \sqrt{2}y'= 0 or y'= 0. Replacing x and y in y= x^2 gives the equation of that curve in the x'y' coordinate system and then rotate around the x' axis.
