# Tagged Questions

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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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### 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 ...
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 ...