Solid of Revolution

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SolidOfRevolutionCylinders

A solid of revolution is a solid enclosing the surface of revolution obtained by rotating a 1-dimensional curve, line, etc. about an axis. A portion of a solid of revolution obtained by cutting via a plane oblique to its base is called an ungula.

To find the volume of a solid of revolution by adding up a sequence of thin cylindrical shells, consider a region bounded above by z=f(x), below by z=g(x), on the left by the line x=a, and on the right by the line x=b. When the region is rotated about the z-axis, the resulting volume is given by

[画像: V=2piint_a^bx[f(x)-g(x)]dx. ]

The following table gives the volumes of various solids of revolution computed using the method of cylinders.

solid f(x) g(x) volume

cone h(1-x/r) 0 1/3pihr^2

conical frustum [画像:{h for x<R_2; h(1-(x-R_2)/(R_1-R_2)) otherwise] 0 1/3pih(R_1^2+R_1R_2+R_2^2)

cylinder h 0 pihr^2

oblate spheroid [画像:csqrt(1-(x^2)/(a^2))] [画像:-csqrt(1-(x^2)/(a^2))] 4/3pia^2c

prolate spheroid [画像:csqrt(1-(x^2)/(a^2))] [画像:-csqrt(1-(x^2)/(a^2))] 4/3pia^2c

sphere sqrt(r^2-x^2) -sqrt(r^2-x^2) 4/3pir^3

torus sqrt(a^2-(x-c)^2) -sqrt(a^2-(x-c)^2) 2pi^2a^2c

spherical segment [画像:{h+d for x<b; sqrt(R^2-x^2) otherwise] d 1/6pih(3a^2+3b^2+h^2)

torispherical dome [画像:{sqrt(R^2-x^2)-(R-h) for x<c[1+(R/a-1)^(-1)]; sqrt(a^2-(x-c)^2) otherwise] 0 pi/6[3a^2cpi+4R^3-2sqrt((a-c-R)(a+c-R))(2a^2+c^2+2aR+2R^2)+6a^2csin^(-1)(c/(a-R))]

SolidOfRevolutionDisks

To find the volume of a solid of revolution by adding up a sequence of thin flat washers, consider a region bounded on the left by x=f(z), on the right by x=g(z), on the bottom by the line z=a, and on the top by the line z=b. When the region is rotated about the z-axis, the resulting volume is

[画像: V=piint_a^b{[f(x)]^2-[g(x)]^2}dx. ]

The following table gives the volumes of various solids of revolution computed using the method of washers.

solid f(x) g(x) volume

barrel (elliptical) [画像:sqrt(r_2^2+((r_1-r_2)(r_1+r_2)(h-2z)^2)/(h^2))] 0 1/3hpi(r_1^2+2r_2^2)

barrel (parabolic) r_2+((r_1-r_2)(h-2z)^2)/(h^2) 0 1/(15)hpi(3r_1^2+4r_1r_2+8r_2^2)

cone r(1-z/h) 0 1/3pihr^2

conical frustum R_1+(R_2-R_1)z/h 0 1/3pih(R_1^2+R_1R_2+R_2^2)

cylinder r 0 pir^2h

oblate spheroid [画像:asqrt(1-(z^2)/(c^2))] 0 4/3pia^2c

prolate spheroid [画像:asqrt(1-(z^2)/(c^2))] 0 4/3pia^2c

sphere sqrt(r^2-z^2) 0 4/3pir^3

torus c+sqrt(a^2-z^2) c-sqrt(a^2-z^2) 2pi^2a^2c

spherical segment sqrt(R^2-z^2) 0 1/6pih(3a^2+3b^2+h^2)

torispherical dome [画像:{c+sqrt(a^2-z^2) for z<a(R-h)/(R-a); sqrt(R^2-[z+(R-h)]^2) otherwise] 0 pi/3[2hR^2-(2a^2+c^2+2aR)(R-h)+3a^2csin^(-1)((R-h)/(R-a))]

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References

Harris, J. W. and Stocker, H. "Solids of Rotation." §4.10 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 111-113, 1998.

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Solid of Revolution

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Weisstein, Eric W. "Solid of Revolution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SolidofRevolution.html

Solid of Revolution

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