Intereting Posts

Question on Inverse Pochhammer Symbol
Volume in higher dimensions
If $x^2 \equiv a \pmod n$, then $x^2 \equiv a \pmod {p_i}$, where $n=p_1^{t_1} \dots p_r^{t_r} $: why?
What is the difference between homomorphism and isomorphism?
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$
Integrating $\int^2_{-2}\frac{x^2}{1+5^x}$
Given this transformation matrix, how do I decompose it into translation, rotation and scale matrices?
How to find the distance between two planes?
Adjoint functors as “conceptual inverses”
What are some real-world uses of Octonions?
Why aren't these loops homotopic?
Stirling number proof, proving that $s(n, n-2) = 2{n\choose3} + 3{n\choose4}$
product of hermitian and unitary matrix
How to use LU decomposition to solve Ax = b
Mean and variance of geometric function using binomial distribution

Show that the surface area of a zone of a sphere that lies

between two parallel planes is $2\pi Rh$,

Where $R$ is the radius of the sphere and $h$ is the distance between the

planes.

If you are wondering what is interesting about this ?

- composition of continuous functions
- How to calculate volume of a cylinder using triple integration in “spherical” co-ordinate system?
- Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
- Asymptotic behavior of the partial sums $\sum\limits_{k=1}^{n}k^{1/4} $
- Monotonocity of the function
- continuous partial derivative implies total differentiable (check)

The fact that the surface area depends only on distance between the planes, and not where they cut the sphere.

I am looking to understand a calculus based solution.

- A limit problem $\lim\limits_{x \to 0}\frac{x\sin(\sin x) - \sin^{2}x}{x^{6}}$
- Number of tangents
- Evaluate the series $\lim\limits_{n \to \infty} \sum\limits_{i=1}^n \frac{n+2}{2(n-1)!}$
- Find the sign of $\int_{0}^{2 \pi}\frac{\sin x}{x} dx$
- Understanding Dirac delta integrals?
- Taking the second derivative of a parametric curve
- Find the Fourier transform of $\frac1{1+t^2}$
- Determining if a quadratic polynomial is always positive
- On the inequality $ \int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$
- Defining the derivative without limits

A sphere is generated by rotating the right portion of a circle centered at the origin and with radius R about the y-axis, that is, by rotating the curve x=√R^2-y^2 about the y-axis, so the surface area of the required zone would be ∫ (from -c to c) 2π*√R^2-y^2*√R^2/R^2-y^2 dy, now we know that 2c=h( assume that c>0), we got h*∫ 2πR dy=2πRh

Consider an “infinitesimal latitude annulus” $A$ of geographical width $\Delta\theta$, positioned at the geographical latitude $\theta\in\left]-{\pi\over2},{\pi\over2}\right[$. Its area is given by

$${\rm area}(A)=2\pi R\cos\theta\cdot R\Delta\theta\ .$$

Now a glance at the “infinitesimal right triangle” with hypotenuse $R\Delta\theta$ reveals that the increment of the $z$-coordinate across $A$ amounts to $\Delta z=\cos\theta\> R\Delta\theta$. It follows that in fact

$$ {\rm area}(A)=2\pi R\>\Delta z\ .$$

You can put it this way: The projection of $S^2$ along horizontal rays onto the cylinder touching $S^2$ along the equator is area-preserving.

- Extension of Sections of Restricted Vector Bundles
- About Cantor's proof of uncountability or real numbers
- Some trigo identities
- Different approaches to N balls and m boxes problem
- Non-invertible elements form an ideal
- The determinant of block triangular matrix as product of determinants of diagonal blocks
- The marriage problem with the constraint that a particular boy has to find a wife.
- Borel $\sigma$ algebra on a topological subspace.
- Exotic Manifolds from the inside
- “Completeness modulo Godel sentences”?
- Find asymptotics of $x(n)$, if $n = x^{x!}$
- Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
- what functions or classes of functions are Riemann non-integrable but Lebesgue integrable
- Convergence of sequence in uniform and box topologies
- Tetrahedral Law of Cosines Proof