I assume this means that the radius goes from one side of the circle to the other for the given value of x which the plane lies on.one of its radii in the base

If the plane lies perpendicular to the x axis then is crosses the x axis at only one point.

For a given point x where the plane crosses the x axis the "height" of the circle is (if you draw a 2D diagram of the circle with the y axis in the upwards direction then it is more clear what I mean by height)

The radius of the quarter circle at this point will be .

The area of the quarter circle will be .

The volume of a small slice of the quarter circle is dV. .

Integrate the volume between the values of x where a plane crosses the circle to get the volume of the solid