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Like utilizing vectors, calculus, differential equations? Is it possible, i am specifically looking at cube or sphere as i know a cone can be easily created by rotating a simple function around the y axis by 2(pi) creating a cone but how would one create a function for a sphere or cube?

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    $\begingroup$ Why couldn't you rotate a circle around the y-axis? $\endgroup$ – Zestylemonzi Nov 27 '16 at 19:48
  • $\begingroup$ You want the domain to be a subset of $\mathbb{R}$? $\endgroup$ – Vitor Borges Nov 27 '16 at 19:55
  • $\begingroup$ How about sweeping out a volume by translating a square in the $xy$-plane along the $z$ axis? $\endgroup$ – user137731 Nov 27 '16 at 20:06
  • $\begingroup$ but how would one do this? $\endgroup$ – Curios and Knowledgeable Nov 27 '16 at 20:07
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A sphere can be defined parametrically as follows:

$$(r \cos θ_1 \cosθ_2, r\cos θ_1 \sinθ_2,r\sin θ_1)$$ for $ 0 ≤ θ_1, θ_2 < 2 π $ where $r$ is the radius of the sphere.

Alternatively, as mentioned by Zestylemonzi, you could rotate the function $ y = \sqrt{r^2 - x^2}$ across the x-axis, again where $r$ is the radius of the sphere.

A cube is a little more tricky as it cannot be described by a single function (to my knowledge). However, the three-dimensional region where $0 ≤ x, y, z ≤ s$, where $s$ is the side length of the cube seems to fit the conditions of the problem.

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You want to generate a surface area.

Like the cone choose a circle centered on y axis and rotate it around y axis to form a sphere.

To form one of 6 faces of a cube length $2$ define that by varying $x,y,z$ as

$$ x=1 \, , (-1<y< 1),\, (-1<z< 1)\,. $$

In Mathematica

ParametricPlot3D[ {1, y, z}, {y, -1, 1}, {z, -1, 1}]
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