Prove that for every constant $c>0$ the set:
$$B_{f, g} = \{ (x, y, z) \in \mathbb{R}^3 : (x-f(z))^2 + (y-g(z))^2 \leq c, \quad z \in [a, b] \}$$
has the same volume for all continuous functions $f, g:[a, b] \rightarrow \mathbb{R}$.
Solution
We are invoking Cavalieri's principal. Since $B$ is bounded by two hyperplanes , let us call them $w=b, \; w=0$ respectively and the intersection of $B$ with any other hyperplane is a Jordan countable set then the volume of $B$ is equal to the sum of the disks that are generated when the hyperplane cuts the surface. However, this sum can be expressed as the integral:
$$V= \int_a^b \pi \sqrt{c}^2 \, {\rm d} z = \pi c (b-a)$$
and we note that the volume is indeed independenant of $f, g$ and dependant of $c$ (as it was supposed to be in any case). Hence the exercises comes to an end!
We give a visualization when $f=g=0$.
Here we note that the plane cuts the cylinder (surface) in circles. Therefore the volume of the cylinder is the sum of the areas of the circles.
$$B_{f, g} = \{ (x, y, z) \in \mathbb{R}^3 : (x-f(z))^2 + (y-g(z))^2 \leq c, \quad z \in [a, b] \}$$
has the same volume for all continuous functions $f, g:[a, b] \rightarrow \mathbb{R}$.
Solution
We are invoking Cavalieri's principal. Since $B$ is bounded by two hyperplanes , let us call them $w=b, \; w=0$ respectively and the intersection of $B$ with any other hyperplane is a Jordan countable set then the volume of $B$ is equal to the sum of the disks that are generated when the hyperplane cuts the surface. However, this sum can be expressed as the integral:
$$V= \int_a^b \pi \sqrt{c}^2 \, {\rm d} z = \pi c (b-a)$$
and we note that the volume is indeed independenant of $f, g$ and dependant of $c$ (as it was supposed to be in any case). Hence the exercises comes to an end!
We give a visualization when $f=g=0$.
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