Ungula
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: Cylindrical ungula. A cylindrical ungula of base radius "r" and height "h" has volume Its total surface area is the surface area of its curved sidewall is and the surface area of its top (slanted roof) is Section::::"Proof". Consider a cylinder formula_5 bounded below by plane formula_6 and above by plane formula_7 where "k" is the slope of the slanted roof: Cutting up the volume into slices parallel to the "y"-axis, then a differential slice, shaped like a triangular prism, has volume where is  the area of a right triangle whose vertices are, formula_11, formula_12, and formula_13, and whose base and height are thereby formula_14 and formula_15, respectively. Then the volume of the whole cylindrical ungula is which equals after substituting formula_19. A differential surface area of the curved side wall is which area belongs to a nearly flat rectangle bounded by vertices formula_21, formula_22, formula_23, and formula_24, and whose width and height are thereby formula_25 and (close enough to) formula_26, respectively. Then the surface area of the wall is where the integral yields formula_28, so that the area of the wall is and  substituting formula_30 yields The base of the cylindrical ungula has the surface area of half a circle of radius "r": formula_32, and the slanted top of the said ungula is a half-ellipse with semi-minor axis of length "r" and semi-major axis of length formula_33, so that its area is and substituting formula_35 yields Note how the surface area of the side wall is related to the volume: such surface area being formula_37, multiplying it by formula_38 gives the volume of a differential half-shell, whose integral is formula_39, the volume. When the slope "k" equals 1 then  such ungula is precisely one eighth of a bicylinder, whose volume is formula_40. One eighth of this is formula_41. Conical ungula. A conical ungula of height "h", base radius "r", and upper flat surface slope "k" (if the semicircular base is at the bottom, on the plane "z" = 0) has volume where is the height of the cone from which the ungula has been cut out, and The surface area of the curved sidewall is As a consistency check, consider what happens when the height of the cone goes to infinity, so that the cone becomes a cylinder in  the limit: so that which results agree with the cylindrical case. Section::::"Proof". Let a cone be described by where "r" and "H" are constants and "z" and "ρ" are variables, with and Let the cone be cut by a plane Substituting this "z" into the cone's equation, and solving for "ρ" yields which for a given value of "θ" is the radial coordinate of the point common to both the plane and the cone that is farthest from the cone's axis along an angle "θ" from the "x"-axis. The cylindrical height coordinate of this point is So along  the direction of angle "θ", a cross-section of the conical ungula looks like the triangle Rotating this triangle by an angle formula_57 about the "z"-axis yields another triangle with formula_58, formula_59, formula_60 substituted for formula_61, formula_62, and formula_63 respectively, where formula_59 and formula_60 are functions of formula_58 instead of formula_61. Since formula_57 is infinitesimal then formula_59 and formula_60 also vary infinitesimally from formula_62 and formula_63, so for purposes of considering the volume of the differential trapezoidal pyramid, they may be considered equal. The differential trapezoidal pyramid has a trapezoidal base with a length at the base (of the cone ) of formula_73, a length at the top of formula_74, and altitude formula_75, so the trapezoid has area An altitude from the trapezoidal base to the point formula_77 has length differentially close to (This is an altitude of one of the side triangles of the trapezoidal pyramid.) The volume of the pyramid is one-third its base area times its altitudinal length, so the volume of the conical ungula is the integral of that: where Substituting the right hand side into the integral and doing some algebraic manipulation yields the formula for volume to be proven. For the sidewall: and the  integral on the rightmost-hand-side simplifies to formula_82. ∎ As a consistency check, consider what happens when "k" goes to infinity; then the conical ungula should become a semi-cone. which is half of the volume of a cone. which is half of the surface area of the curved wall of a cone. Surface area of top part. When formula_86, the "top part" (i.e., the flat face that is not semicircular like the base) has a parabolic shape and its surface area is When formula_88 then the top part has an elliptic shape (i.e., it is less than one -half of an ellipse) and its surface area is where When formula_94 then the top part is a section of a hyperbola and its surface area is where