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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 .
Cylinder: A cylinder can be expressed a solid of revolution by revolving the line around the -axis. To find the volume, we can find the area under the curve, and then when we revolve it, it becomes the volume. The radius is and the height, , is the upper bound of integration. We have . Integrating, we get . This is the formula of a cylinder.
Cone: If you are given the height and radius of the cone, and you have the point on your line(since the vertex is 0), then , because the height is the x-coordinate and the radius is the y(for the same reason seen above in the cylinder). Now, since we have , we know the y-intercept, and we can only have one slope. If , and is the slope, then we have , and therefore , so the equation is . For the integral, we get .
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 . The diameter is to , so that is where we integrate. .