Tagged Questions

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A little help integrating this torus?

Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by $$\mathbf{F}(x,y,z)=(x,y,z).$$ Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
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How to integrate $\cos\left(\sqrt{x^2 + y^2}\right)$

Could you help me solve this? $$\iint_{M}\!\cos\left(\sqrt{x^2+y^2}\right)\,dxdy;$$ $M: \frac{\pi^2}{4}\leq x^2+y^2\leq 4\pi^2$ I know that the region would look like this and I need to solve it as ...
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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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Calculate volume in a 3D sort of space using cartesian coordinates

Find the volume bounded by the cylinder $x^2 + y^2 = 1$, the planes $x=0, z=0, z=y$ and lies in the first octant. (where x, y, and z are all positive)
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Change of variables in two dimensions

This is from Munkres' Analysis on Manifolds, Section 17, Question 4. (a) Show that $$\int_\Bbb {R^2} e^{-(x^2+y^2)} = \left[ \int_\Bbb R e^{-x^2}\right]^2,$$ provided the first of these ...
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Riemann integrable then J-integrable

Let $E\subset\mathbb{R}^n$ be a closed Jordan domain and $f:E\rightarrow\mathbb{R}$ a bounded function. We adopt the convention that $f$ is extended to $\mathbb{R}^n\setminus E$ by $0$. Let $\jmath$ ...
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Riemann integral is zero for certain sets

The question is: Let $\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}$. Prove that if $E\subset\pi$ is a closed Jordan domain, and $f:E\rightarrow\mathbb{R}$ is Riemann integrable, ...
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help taking line integral over a vector field

I have a problem in which I'm given a force field $\vec{F}(x,y,z)=x\hat{i}+y\hat{j}+ 3\hat{k}$ and a path $\vec{r}(t)=4cos(t)\hat{i}+4sin(t)\hat{j}+3t\hat{k}$ over the interval $0\le t\le 2\pi$. I ...
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Line and surface integrals $R^{3}$

So i actually missed the class where this material was covered so plaese bear with me if my understanding is not so good. one of the problems in my textbook is as follow's. Prove the following ...
I've been revising this area and I've completely forgotten what I'm doing and my notes are sketchy. Evaluate $\int r \cdot dS$ over the surface of the sphere, radius a, centred at the origin. ...
I was given this question in class and I was just wondering if I am on the right trackā¦ Evaluate: $$I=\iint\left(1-\frac{x^2}{a^2} -\frac{y^2}{b^2} \right)^{3/2} dxdy$$ over the region enclosed by ...