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[hide]My Solutions
Some Proofs I wrote
if is prime.
Proof: Expanding out, all the coefficients are of the form by the binomial theorem. To prove the original result we must show that if and , then . Because , , which is divisible by , so the original expression must be divisible by . However if is prime, , since does not contain (because ). Therefore, in order for to be divisible by , is divisible by . All the coefficients of the expansion(besides the coefficients of and ) are of the form , and , so they cancel out and if is prime.
Volume of Cylinder, Cone, and Sphere
If we have a function , that can be rotated to make a shape, the area underneath it will turn into the volume. However, since we are revolving it in a circular motion, the area will actually become the radius. Another way of seeing this is splitting it into infinite circles and adding up all of them. Therefore, for a function , we have the volume of the solid of revolution to be $\pi <cmath> \int_{a}^{b} (f(x))^2 \,dx </cmath>.
Cylinder: A cylinder can be expressed a solid of revolution by revolving the line$ (Error compiling LaTeX. Unknown error_msg)y = rxrh$, is the upper bound of integration. We have <cmath>\pi <cmath> \int_{0}^{h} r^2 \,dx </cmath></cmath>. Integrating, we get <cmath>\pi (r^2h - r^2(0)) = \pi r^2h</cmath>. This is the formula of a cylinder.
Cone: If you are given the height and radius of the cone, and you have the point (0,0) on your line(since the vertex is 0), then$ (Error compiling LaTeX. Unknown error_msg)f(h) = r(0,0)h=xmr = mhm = \frac{r}{h}f(x) = \frac{rx}{h}$. For the integral, we get <cmath>\pi [ \int_{0}^{h} \frac{r^2x^2}{h^2} \,dx \] = \pi \frac{r^2h}{3}</cmath>.
Sphere: The equation of a sphere should be a circle, but that is a relation and not a function. Therefore, we can use the top half of a circle, and the bottom half will get filled in when it rotates. Therefore, we get$ (Error compiling LaTeX. Unknown error_msg)f(x) = \sqrt{r^2 - x^2}-rr$, so that is where we integrate.
\[\pi [ \int_{-r}^{r} r^2 - x^2 \,dx \] = \frac{4\pi r^3}{3}\] (Error compiling LaTeX. Unknown error_msg)
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