# Tagged Questions

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### Partial derivative of straigh-line parametrized integral

I would like to evaluate the following $$F(\mathbf{r}_1,\mathbf{r}_2) = \int_0^1 ds~f(\mathbf{r}_1 + (\mathbf{r}_2 - \mathbf{r}_1)s)$$ where $\mathbf{r}_{1/2} = (x_{1/2} , y_{1/2})$, i. e. a ...
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### Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
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### Prove the continuity and differentiability of parametric integration

$$F(\alpha )=\int_{0}^{+ \infty } \frac{\cos x}{1+(x+\alpha )^{2} } dx$$ Prove the function F is continuous and differentiable on the interval $[0, +\infty )$
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### Calculate the convergence domain of parameter improper integral

$$\int^{+ \infty }_{1} x^{u} \frac{x + \sin x}{x - \sin x}dx$$ The answer is $u<-1$. I suppose we need to find the simplified equivalent form of it, but I stuck on my way.
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### area under a parametric curve problem

so I have a parametric curve, x = cos(t) y = sin(2t) I found that I need the area from 0 to pi/2. put this into an integral in terms of t I get $$-\int_0^{\pi/2}sin(2t)sin(t)dt$$ But in my ...
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### Some questions about parametric integrals

1) What is the error in the following calculation ? $\int_{0}^{oo} \frac {sin(px)}{x}dx$=$\frac {\pi}{2}$ derivating by p at both sides $\int_{0}^{oo} cos(px)dx$=0 But the second integral does not ...
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### Using geometric arguments to solve an analysis problem

Im not good in geometric interpretations... any help is very welcome. Consider the unitary disc $$D=\{(x,y,0)\in\mathbb{R}^3, x^2+y^2\leq1\},$$ parameterized by ...
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### Parametric representation of $\sqrt{x^2+y^2}\le z \le 2$

just wondering how to parametrize this. Question is: Let $C$ denote the conical region $\sqrt{x^2+y^2}\le z \le 2$. Find a parametric representation $\mathbf{x}(u,v)$ for $S$, the surface of $C$. ...
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### Explanation of the area under the curve given by a parametric equation

My textbook says the area under a graph is given by: $\smallint ydx$ And it then goes on to say by the chain rule: $$\smallint ydx = \smallint y{{dx} \over {dt}}dt$$ Could someone explain to me how ...
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### Finding surface area of a cone

I will describe the problem then show what I tried to solve it. I need to find the area of the cone defined as follows: $$z^2=a^2(x^2+y^2)$$ $$0\leq z\leq bx+c$$ where $a,b,c>0$ and $b<a$. ...
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### How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 )$. Evaluate the integral : $$\iint_D e^{\frac{y-x}{y+x}}$$ a) by transforming to polar coordinates b) by using the ...
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### Having trouble solving question involving parametric equations

I have been given the following: $$y = a \cdot \cos^3t$$ $$x = a \cdot \sin^3t$$ $$0 \leqslant t \leqslant {\frac\pi2}$$ I am supposed to show that the mean value of $y$ over the interval ...
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### Line integrals of vector fields

Consider the vector field:$$\vec G = \left(\frac y{x^2+y^2}, \frac {-x}{x^2+y^2}\right)$$ compute $\int_\Gamma \vec G$ where $\Gamma$ is the proportion of a parabola $y=a(x-1)^2$ from (1,0) to (2,a). ...
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### Computing the surface area of a (piecewise) polynomial parametric surface

I'm wondering what kind of numerical integration (e.g. Gauss-Legendre quadrature) I should use to compute the surface area of a (piecewise) polynomial parametric surface. There are two cases. Case ...
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### parametric curves, parameter and integration

I just started learning about parametric curves and I find it confusing that we have a 3rd variable but this 3rd variable "t" is some imaginary variable....I dont get what the difference is between ...
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### Help me understand a surface integral question?

The question is: Evaluate the surface integral: $$\iint\limits_S \, x^2yz\ \mathrm{d} S$$ Where S is part of the plane z = 1 + 2x + 3y that lies above the rectangle [0,3] X [0,2] I literally just ...
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### Finding parametric equations

I am trying to understand volume and surface integrals. I do get the idea of the process (find a parametric equation of the volume/surface, integrate afterwards). But I just cannot make up parametric ...