multivariable calculus

Impossible Integral?

The following integral may be problematic for a freshman calculus student, even if armed with a list of antiderivatives: INTEGRAL0 to infinity exp(-x2) dx. Why? Well, there isn’t a closed-form expression for the antiderivative of the integrand, so the Fundamental Theorem of Calculus won’t help. But...

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Polyhedral Face Vectors

A polyhedron is any closed region of 3-space cut out by a finite set of planes. Take any polyhedron and do the following: on each face, place a vector perpendicular to that face with length proportional to the area of that face. No matter what polyhedron you started...

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Spherical Bands

Take any sphere and slice it up by parallel knives 1 inch apart. The areas of the spherical bands between the cuts (on the surface of the sphere) are exactly the same area! Presentation Suggestions:Draw a suggestive picture! The Math Behind the Fact:This begins to make sense when you reflect on it....

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Right Double Cone

Take a solid cylinder of radius R and height 2R. Remove the right “double cone” which passes through the center of the solid cylinder and extends meet the circular discs on the top and bottom of the cylinder. See Figure 1. The volume of the...

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