# Tagged Questions

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### Normal Vector Affecting The Divergence Theorem

I'm going to use an example to explain what I'm trying to ask. Let $T =$ {$(x,y,z): x^2+y^2=z^2, 0\leq z\leq3$}, $F=(P,Q,R)$ vector field. I'm asked to calculate $\iint_T F$ using the outer pointing ...
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### line integrals and partial derivatives statement (Green's theorem application)

Let $P(x,y),Q(x,y)$ be $C^1$ functions of $\mathbb R^2$, prove that the following statements are equivalent: (1) $P_x-Q_y=0$ and $P_y+Q_x=0$ (2) For every simple closed curve $C$, it is satisfied ...
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### Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
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### Integral of a bivariate normal cdf

Let $$\Phi_2(x,y;\rho):=\int_{-\infty}^y\int_{-\infty}^x \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{1}{2(1-\rho^2)}(s^2+t^2-2st\rho)} \, ds \, dt$$ be the joint cdf of bi-variate normal random ...
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### Find the work done by the force field in moving the particle from one point to another

Find work done by the force field F in moving the particle from $(-1, 1)$ to $(3, 2)$ This sounds good till we are given that $\textbf{F} = \dfrac{2x}{y}\textbf{ i }- \dfrac{x^2}{y^2}\textbf{ j }$ ...
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### Calculating moment of inertia about the $z$-axis of solid with constant density

I have the following math problem: Find the moment of inertia about the $z$-axis of the solid in the first octant that is bounded by the coordinate planes and the graph of $x+y+z=1$ if the density ...
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### how to increace the volume to a specific volume in revolution of solid, using integration [closed]

two functions, f(x)= 1/9(x-2)^2+7 domain range:{0,10}, g(x)=1/7(x-5)+0.7 domain range: {10,13} increase the volume to 1000ml to 1050mL using integration.
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### Changing order of integration (multiple integral)

Prove $$\int_0^a\left( \int_0^x \left( \int_0^y \left( \int_0^z f(u) \, du \right) dz \right) dy \right) dx = \int_0^a \frac {(a-t)^3}{3!} f(t) dt$$ where $a$ is constant. So I began with two ...
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### Find $\iiint_E sin^3 x+\tan y+ 6\hspace{1mm} dV$, where $V$ is region inside $x^2+y^2+z^2 = 1$

I guess that the integral of $\sin^3 x+\tan x$ part is zero, because i have seen many problems like these where the integral is over a symmetrical region and the functions are odd. But I want ...
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### Finding the partial derivatives of $h(x)=\int_{0}^{\|x\|} f(t)\, dt$

Find the partial derivatives of $$h(x_1,\dots,x_n)=\int_{0}^{\|x\|} f(t) dt$$ where $\|x\|$ is the Euclidean norm of $x=(x_1,\dots,x_n)$ and $f$ is some continuous function. I'm sorry but I'm really ...
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### Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
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### Area of the region: $\;x ≥ 0; \;−x\sqrt3 ≤ y ≤ x\sqrt3;\,\;(x−1)^2 + y^2 ≤ 1$.

Can anyone please explain how to set up the needed integral? I need to calculate the area of the region given by: $x ≥ 0,$ $-x\sqrt3 ≤ y ≤ x\sqrt3,$ $(x−1)^2 + y^2 ≤ 1$.
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### What is the meaning of $d\vec S$ in a surface integral?

Can someone explain if I have a surface $z= 9-x^2-y^2$ What would $\vec{n}$ be? What would $d\vec{S}$ be? Why is $d\vec{S}$ $(2x,2y,1)$ and not $(2x,2y,1)/\sqrt{4x^2+4y^2+1}$? Thanks!
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### Find finite area between curves [closed]

Find the finite area enclosed between $r= a \sin 4(\theta)$ and $r= a \sin 2(\theta)$ in polar coordinate system.
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### Using Stokes theorem to find the integral of a vector field over the curve of intersection of two surfaces

Find $\int_C{ \vec{F} \cdot \vec{dr}},$ where $F(x, y, z) = \langle 2 x^2 y , 2 x^3 /3, 2xy\rangle$ and $C$ is the curve of intersection of the hyperbolic paraboloid $z = y^2 - x^2$ and the ...
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### Find $\int_0^1 \int_{3x}^3 (x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$ [closed]

You can use a calculator after simplification if its not possible by hand All Ideas will be appreciated Also If you could find $$\int_0^1 \int_{3x}^3 x(x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$$ ...
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### multivariable calculus double integration volume question

Use a double integral to find the volume of the solid bounded by graphs of the equations given by: \begin{align}z=xy^2, \text{ where: } &z>0\\&x>0\\&5x<y<2\end{align} My ...
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### Find $\iiint_E (1-x^2-2y^2-3z^2)~\mathrm{d}V$, where $E$ is the region inside the ellipsoid $x^2+2y^2+3z^2=1$ [closed]

My textbook asked to use a calculator to find this. Not sure how to setup the triple Integral.
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### find $\int_0^4\int_0^4\int_0^4 \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz$

I am looking for an approximation to the nearest integer of $$\int_0^4\int_0^4\int_0^4 \sqrt{x^2+y^2+z^2}\,dx\,dy\,dz.$$ Wolfram alpha gives up and says "computation time exceeded". I tried, ...
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### Evaluating a path integral without being given a parametrization

Find the mass of the wire formed by the intersection of the sphere $x^2 + y^2 + z^2 = 1$ and the plane $x + z = 0$ if the density of the wire is $6y^2$ grams per unit length. I am completely stuck on ...
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### Calculations of double integral of $xy^2$ over the region between $y=0$ and $y=4-x^2$

Find $\iint_D f~\mathrm{d}A$ where $f(x,y) = xy^2$. Region $D$ is between $y=0$ and $y=4-x^2$ If you draw the graph, you can see that it equal to $0$, however if you calculate the answer ...
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### Help with calculating line integrals and potential functions [duplicate]

May you please help me with this questions? 1) Among all smooth, simple closed curves in the plan, oriented counterclockwise, find one along which the work done by the following vector is greatest: ...
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### potential functions and line integrals

May you please help me with this questions? 1) Among all smooth, simple closed curves in the plan, oriented counterclockwise, find one along which the work done by the following vector is greatest: ...
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### line integral…

Calculate $$\int_Γ f \, d\ell$$ for $f(x,y) = y, \; y=x^{1/2}$, $x$ is in $[2,6]$. I know (now) that it means that: $$\int_\Gamma f \, d\ell=\int_a^b f(\Gamma(t)) \cdot \|\dot\Gamma(t)\| \, dt$$ ...
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### Simplification of an integral comprising of vector-variables

How can I evaluate the simplify the integral $\int \rho (\bf{r^{\prime}})\, \delta (\frac{\sigma}{2}-r) d \bf{r^{\prime}}$ where $\delta$ is the dirac-delta function given that $\rho$ is constant ...
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### How to treat Dirac delta function of two variable?

We can treat one variable delta function as $$\delta(f(x)) = \sum_i\frac{1}{|\frac{df}{dx}|_{x=x_i}} \delta(x-x_i).$$ Then how do we treat two variable delta function, such as $\delta(f(x,y))$? ...
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### Need Help Understanding How To Integrate With An Implicit Variable

My calculus is really rusty (damn Mathematica/Matlab) and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper I am trying ...
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### Double integral where limits are the first quadrant

Evaluate the integral $$\iint\limits_D \frac{1}{(x+y+1)^3} \, dA$$ where $D$ is the first quadrant. In this case, what would the limits of integration be? I'm having trouble moving to polar ...
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### Work done by a force field line integrals

Find the work done by the force field $F(x, y) = \langle 2x \sin(y), 2y \rangle$ on a particle that moves along the parabola $y = x^2$ from $(-1, 1)$ to $(2, 4)$. So to use line integrals to solve ...
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### Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right)$ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
The vector field $F(x, y) = \left(\displaystyle\frac{x}{r^3}, \frac{y}{r^3}\right)$ appears in electrostatics, where $r = \sqrt{x^2 + y^2}$ is the distance to the charge. Find a function $f(x, y)$ ...