In solid geometry, an ungula is a region of a solid of revolution, cut off by a plane oblique to its base. A common instance is the spherical wedge. The term ungula refers to the hoof of a horse, an anatomical feature that defines a class of mammals called ungulates. The volume of an ungula of a cylinder was calculated by Grégoire de Saint Vincent. Two cylinders with equal radii and perpendicular axes intersect in four double ungulae. The bicylinder formed by the intersection had been measured by Archimedes in The Method of Mechanical Theorems, but the manuscript was lost until 1906. A historian of calculus described the role of the ungula in integral calculus:
Grégoire himself was primarily concerned to illustrate by reference to the ungula that volumetric integration could be reduced, through the ductus in planum, to a consideration of geometric relations between the lies of plane figures. The ungula, however, proved a valuable source of inspiration for those who followed him, and who saw in it a means of representing and transforming integrals in many ingenious ways.
Cylindrical ungula
A cylindrical ungula of base radius r and height h has volume
V = 2 3 r 2 h {\displaystyle V={2 \over 3}r^{2}h} ,. Its total surface area is
A = 1 2 π r 2 + 1 2 π r r 2 + h 2 + 2 r h {\displaystyle A={1 \over 2}\pi r^{2}+{1 \over 2}\pi r{\sqrt {r^{2}+h^{2}}}+2rh} , the surface area of its curved sidewall is
A s = 2 r h {\displaystyle A_{s}=2rh} , and the surface area of its top (slanted roof) is
A t = 1 2 π r r 2 + h 2 {\displaystyle A_{t}={1 \over 2}\pi r{\sqrt {r^{2}+h^{2}}}} .
Proof Consider a cylinder x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} bounded below by plane z = 0 {\displaystyle z=0} and above by plane z = k y {\displaystyle z=ky} where k is the slope of the slanted roof:
k = h r {\displaystyle k={h \over r}} . Cutting up the volume into slices parallel to the y-axis, then a differential slice, shaped like a triangular prism, has volume
A ( x ) d x {\displaystyle A(x)\,dx}
where
A ( x ) = 1 2 r 2 − x 2 ⋅ k r 2 − x 2 = 1 2 k ( r 2 − x 2 ) {\displaystyle A(x)={1 \over 2}{\sqrt {r^{2}-x^{2}}}\cdot k{\sqrt {r^{2}-x^{2}}}={1 \over 2}k(r^{2}-x^{2})}
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