Tagged Questions

21
votes
4answers
496 views

The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
0
votes
0answers
60 views

Calculate the total charge within each of the indicated volumes: engineering electromagnetic [closed]

Calculate the total charge within each of the indicated volumes: 0.1 ≤│x│, │y│, │z│ ≤ 0.2: pv= 1/ x^3 y^3 z^3; 0 ≤p ≤ 0.1, 0≤ Ø≤pi, 2 ≤ z ≤ 4; pv=p^2 z^2 sin 0.6 Ø; universe: pv= e^-2r/r^2 ans: ...
5
votes
2answers
171 views

Math Courses involving clever integration techniques

I am a third year undergraduate mathematics student. I learned some basic techniques for simplifying sums in high school algebra, but I have encountered some of the more interesting techniques in my ...
3
votes
3answers
58 views

Moment, spheroid, charge redistribution

Let $$I_k:= c \int_{\mathbb R^3} (3x_k'^2-r'^2) \,\,\,d^3 x'$$ where ${r'}^2={x'}_1^2+{x'}_2^2+{x'}_3^2$ and $c$ is a constant = density of charge (uniform) in the body. Suppose this integral is ...
2
votes
1answer
150 views

Showing that an integral can not be expressed in terms of elementary functions

I recently encountered an integral of the form: $$\int{\frac{\log(a+bx+\sqrt{x^2+c})}{x}}dx$$ The result involves the dilogarithm function, but I was wondering if there is a fast way of showing that ...
10
votes
1answer
206 views

How can I intuit the role of the central limit theorem in breaking the curse of dimensionality for Monte Carlo integration

I would like to more intuitively understand where the power of Monte Carlo integration comes from for large-dimensional domains of integration. Other questions on this site have referenced the proof ...
2
votes
3answers
143 views

Can we possibly combine $\int_a^b{g(x)dx}$ plus $\int_c^d{h(x)dx}$ into $\int_e^f{j(x)dx}$?

I'm wondering if this is possible for the general case. In other words, I'd like to take $$\int_a^b{g(x)dx} + \int_c^d{h(x)dx} = \int_e^f{j(x)dx}$$ and determine $e$, $f$, and $j(x)$ from the other ...
4
votes
2answers
253 views

Why is this constant of integration taken as $\log A$ instead of just $C$?

Suppose we solve $$\frac{dy}{dx} = \frac{1 + y}{2 + x} .$$ Which can be written as the following and integrating both sides w.r.t. $y$ and $x$: $$\int\frac{1}{1 + y}dy = \int\frac{1}{2 +x}dx ,$$ we ...
3
votes
4answers
187 views

How to think about derivatives in an abstract fashion?

Derivatives seem easy to understand abstractly as the rate of change of something, higher order derivatives are the rate of change of the rate of change of something, and so on. I, however, have ...
3
votes
3answers
349 views

Why is the area under the curve exponentially greater than the original function?

So I've been a calculus student now for about two years, and I've gone as high as differential equations, but I am still a bit puzzled by the fact that the area under the curve of some function is ...
5
votes
1answer
362 views

A way to see that $\int_{0}^{\infty}\exp(-x)dx=1$?

One can easily find the integral $\int_{0}^{\infty}\exp(-x)dx$. It is equal to 1. But is there a way to understand this geometrically without integration? If i rotate the picture i see that ...
3
votes
1answer
667 views

Connection between chain rule, u-substitution and Riemann-Stieltjes integral

I think I understand these concepts ok: chain rule u-substitution Riemann-Stieltjes integral But there seems to be a layer that I miss: They all seem to be connected, alas I don't know how ...
17
votes
3answers
7k views

Why is the area under a curve the integral?

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ...
2
votes
1answer
337 views

Insidious exponential integral

I hope that someone's up for the challenge; I'm attempting to solve this via computer: \begin{equation} \int_{-\pi}^\pi{\displaystyle \frac{e^{i\cdot a\cdot t}(e^{i\cdot b\cdot t}-1)(e^{i\cdot c ...
2
votes
2answers
363 views

Is this integration approximation method known/used?

I'm approximating an integral with only exponentials. i.e., it is equal to $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{c_j e^{i\cdot d_j \cdot t}}}{\displaystyle\sum_{k=a}^b{r_k ...
1
vote
0answers
274 views

How effective is this alternative to integration?

I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following: $f(x) = ...