# Tagged Questions

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### From 2 to 3 dimensions: integrating a force along a contour/surface.

I am studying the following problem: Consider a closed contour $\mathcal{C}$ in $\mathbb{R}^2$ defined by $r(\theta)$ where $\theta\in[0,2\pi)$ and $r(0)=r(2\pi)$ (let the center to be zero for ...
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### Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
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### Proof that $\int \frac{1}{x}$ is $\ln(x)$

When I was learning Calculus AB and Calculus II/III at my high school, I noticed that our textbooks never gave a full fundamental proof that $\int \frac{1}{x}$ is $\ln(x)$, and rather said that when ...
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### expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
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The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
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### Show $g(\mathbf{x}) \leq h(\mathbf{x})$ implies $\int g(\mathbf{x})\mathrm{d}\mathbf{x} \leq \int h(\mathbf{x})\mathrm{d}\mathbf{x}$

Suppose I have $g$ and $h$ from $\mathbb{R}^p\to\mathbb{R}$ such that for all $\mathbf{x}$, $g(\mathbf{x}) \leq h(\mathbf{x})$. I want to prove that the integral over all $\mathbb{R}^p$ of $g$ is less ...
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### proof that a function is integrable on a interval $[a,b]$

a) Divide a interval $[a,b]$ into $n$ equal subintervals. here I'm thinking $P_{n} =(x_0,x_1,x_2,x_3,x_{n-1}, x_n)$ where $a = x_0 < x_1 < x_2 < x_3 <\dots< x_{n-1} < x_n = b$ ...
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### Proof strategy for Pointwise converging sequence of Riemann integrable functions to not uniformaly converge

I am wondering of a proof strategy to show. That a sequence of Riemann integrable functions which converges point wise to a function may not actually uniformly converge to it. If it makes the argument ...
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### Integration Problem Proof ($\sin x$)

Problem: Integration of $\displaystyle\int_{-1}^1 {\sin x\over 1+x^2} \; dx = 0$ (according to WolframAlpha Definite Integral Calculator) But I don't understand how. I tried to prove using ...
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### Integrating $\int \sin^n{x} \ dx$

I am working on trying to solve this problem: Prove: $\int \sin^n{x} \ dx = -\frac{1}{n} \cos{x} \cdot x \ \sin^{n - 1}{x} + \frac{n - 1}{n} \int \sin^{n - 2}{x} \ dx$ Here are the steps that I ...
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### Representation of an equality

I know that I keep asking the similar problems with a little modification but it is really important to me to make sure that I am at the right track. This is my previous question link. Since we can ...
Prove that $\sum_{n=2}^{\infty} \frac{z^{n-1}}{\alpha(n-1)+1}$ is equivalent to $\frac{1}{\alpha} \displaystyle \int_{0}^{1}{ \frac{z t^{\frac{1}{\alpha}}}{1-tz}} dt$?