0
votes
1answer
30 views

Find average stream velocity over a cross-section of a cylinder, given $v(r) = k(R^2 − r^2)$

I'm doing integration homework and a question that I am not sure how to approach popped up. The velocity (in centimeters per second) of blood r cm from the central axis of an artery is given by ...
0
votes
0answers
36 views

What is this called specifically?

Imagine you take a radius from the center of the shape, you add up all of the lines as it rotates 360 degrees. The radius is measured from its point of rotation, like (0,0) in Cartesian coordinates,to ...
0
votes
1answer
18 views

Average value for multiple integrals

If there is a function $f(x,y)$ and we want to find the average value over a region $R$ defined by $0<x<1$ and $0<y<x$, how is that computed? I know that it would be something like this: ...
2
votes
1answer
24 views

Problem about average of cos square (nt) where n is arbitrary

I often see people just say time average of cos^2(nwt) is 1/2, I want to know in what cases this is not valid? w is just the frequency, can be assumed as a constant. Assuming you are always ...
0
votes
0answers
77 views

How to prove the integral function with cosine is increasing

Prove that the following function is an increasing function on $x\in (0,1)$ when $n\ge2$. $f(x) = \int_0^{\pi} \frac{1}{(1-2x\cos\theta+x^2)^n} d\theta$
2
votes
1answer
39 views

Are the $L^p$ norms ordered by $p$?

A question left over from this post is: Are the $L^p$ norms ordered by $p$ like the power means are?
7
votes
2answers
170 views

What does the $L^p$ norm tend to as $p\to 0$?

This is something I was thinking about, so I'm going to post it as a question and post my own answer. I hope that anyone who wants will comment, correct me if I'm wrong, and add their own knowledge ...
0
votes
0answers
91 views

Average of an exponential over Dirichlet probability distribution on the (n-1)-simplex

Any idea how I can solve this integral for arbitrary integer $n$ ($n \geq 2$) with real non-negative coefficients $\{s_i\}_{i=1,..,n}$: $I(s_1,...,s_n):= \int_0^1 dx_1 ...\int_0^1 dx_{n}\,\, e^{ ...
5
votes
1answer
305 views

Average sine of an angle between two rays in a cone

I'm looking for an average value of sine of an angle between two rays, lying within a cone with a certain angle. Given a cone with an aperture of ${2\chi}$ and two rays lying within the cone. The ...
4
votes
1answer
179 views

Power Mean Random Distribution

I'm trying to find a the distribution for the power mean of $n$ random variables on $[0,1]$. I've got the arithmetic mean: $\frac{n}{(n-1)!}\sum_{k=0}^{\lfloor ...
23
votes
4answers
1k views

What is to geometric mean as integration is to arithmetic mean?

The arithmetic mean of $y_i ... y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$ For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using ...