general
limit of an integral of a sequence of functions
Suppose that $f$ is continuous on $[0,1]$. ($f'(x)$ may or may not exist). How can I show that $$\lim_{n\rightarrow\infty} \int\limits_0^1 \frac{nf(x...
Apr 13, 2026
Archive: General
general
Deriving the Volume of a Cone using Solids of Revolution
Related is Why is the volume of a cone one third of the volume of a cylinder?, but it does not outline finding the volume of a cone using solids of revolu...
Apr 13, 2026
general
How to justify this rule stated by Symbolab Calculator : $f(x)^{g(x)} = e^{g(x)\times ln(f(x))}$
The rule Symbolab Calculator uses to solve $\sqrt[x]{x^3} = 100$ is not familiar to me and I do not think I have already seen it featuring in any usual ex...
Apr 13, 2026
general
Correct name of an irregular pentagon with four vertices in a square shape and one protruding perpendicular from the mid point of two of the vertices?
What is the correct name of an irregular pentagon that has 4 vertices in a square shape and one protruding perpendicular from the mid point of two of the ...
Apr 13, 2026
general
Why are polynomials interesting?
I'm learning polynomials (for competitions) the first time, with having a little number theory and combinatorics experience. The difficulty is that I...
Apr 13, 2026
general
Is the permutation matrix P of PLU decomposition unique?
Let $A$ be a square matrix. Then there exists a permutation matrix $P$ such that $A=PLU$, where $L$ is a lower triangular matrix and $U$ is an upper trian...
Apr 13, 2026
general
How many spanning subgraph of a graph G
If I have a graph $\mathbb G$ with $n$ vertices, $m$ edges and $c$ components, how can I count how many spanning subgraphs it has? Thanks!
Apr 13, 2026
general
Integrals such as x+y wrt x
This question is in lieu of "Integrating $dψ=(x+y)dx+x_0dy$". Though I got good answers, none of them could explain the real question. Consider the integr...
Apr 13, 2026
general
Congruence of triangles: SSA criteria
It is well known that this criteria does not work in general. I am trying to answer to the following question if two triangles have two sides and the angl...
Apr 13, 2026
general
Euclidean Algorithm for polynomials
I know how to use the extended euclidean algorithm for finding the GCD of integers but not polynomials. I can't really find any good explanations of ...
Apr 13, 2026
