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The final step is to then set up the integral for the volume and evaluate it.

For the limits on the integral we can see that the “first” disk in the solid would occur at and the “last” disk would occur at .  Our limits are then : .

The volume is then,

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The area of the disk is then,

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Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by

,

and the y-axis about the y-axis.

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in the real world you will need to know how to find the volume of cylinders in order to build a wooden cup or anything circular

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When a curve is defined by its parametric form (x(t),y(t)) in some interval [a,b], the volumes of the solids generated by revolving the curve around the x-axis or the y-axis are given by[1]Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the x-axis or the y-axis are given by[2]

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The cylinder method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.

The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the y-axis is given by

If g(x) = 0 (e.g. revolving an area between curve and y-axis), this reduces to:

The method can be visualized by considering a thin vertical rectangle at x with height f(x) − g(x), and revolving it about the y-axis; it forms a cylindrical shell. The lateral surface area of a cylinder is 2πrh, where r is the radius (in this case x), and h is the height (in this case f(x) − g(x)). Summing up all of the surface areas along the interval gives the total volume.

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The disk method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution.

The volume of the solid formed by rotating the area between the curves of f(x) and g(x) and the lines x = a and x = b about the x-axis is given byIf g(x) = 0 (e.g. revolving an area between the curve and the x-axis), this reduces to:
The method can be visualized by considering a thin horizontal rectangle at y between f(y) on top and g(y) on the bottom, and revolving it about the y-axis; it forms a ring (or disc in the case that g(y) = 0), with outer radius f(y) and inner radius g(y). The area of a ring is π(R ² − r ²), where R is the outer radius (in this case f(y)), and r is the inner radius (in this case g(y)). The volume of each infinitesimal disc is therefore πf(y)² dy. The limit of the Riemann sum of the volumes of the discs between a and b becomes integral (1).

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Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness δx, or a cylindrical shell of width δx; and then find the limiting sum of these volumes as δx approaches 0, a value which may be found by evaluating a suitable integral.

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a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis of revolution) that lies on the same plane.