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Disc integration

Consider the horn shape created by rotating the region bounded by the line x = 0, the line , and the curve in the x-y-plane, around the x-axis into the z-dimension.[1] We can divide this shape into disks. The area of each disk is , which is in this context. The thickness of the disk is dx, so in three dimensions the volume of the disk is . Adding up all the disks as x changes, we come out with

where this integral has been solved by the method of substitution setting so and the bounds change from 0 and to 0 and .


  1. ^ The example is taken from Calculus: Early Transcendentals, 2nd Edition, by William Briggs, Lyle Cochran, Bernard Gillett & Eric Schulz, p. 424.