# Questions tagged [multiple-integral]

For questions regarding computation and results related to integrals in at least 2 variables.

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### Calculation of Surface Integration.

I've been studying surface integration by myself, but I'm always stuck at the last step. Consider the above question: This is my approach: Calculation of the curl of the given field. Calculation of ...
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### Triple integral, how to solve it? [closed]

I have to calculate $\int_D z^2 dx dy dz$, where $D = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 \le 3, x^2+y^2 \le 2z \}$. I think that a good idea would be to use cylindrical coordinates, but how ...
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### Finding the angular momentum of a spinning ring using an integral

This problem requires you to compute components of the moment of inertia tensor for several simple shapes. In each case, the object will be spinning about its centre of mass with angular speed $\omega$...
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### Computing or upper bounding a complicated integral

I am stuck in trying to compute or more realistically upper bound the following double integral $$\int_\mathbb{R}\int_\mathbb{R} \frac{\exp(-(x-a)^2)\exp(-(y-a)^2)}{1+\exp(-x^2)+\exp(-y^2)}dxdy$$ as ...
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### Algebraic manipulation of the product of multiple differentials

I want to correct some of my misunderstandings in multivariable differentiation from a purely algebraic and non-geometrical viewpoint (assuming one does not know anything about geometry and no ...
1 vote
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### Calculate the volume of the body defined by $x^2+y^2+z^2≤1$ for which $x^2+z^2≥z$

Calculate the volume of the body defined by $x^2+y^2+z^2≤1$ for which $x^2+z^2≥z$ Attempt: For the volume, I used this formula: $V=\iint y(x,z)dxdz$ I find it easier to express the problem through $y$ ...
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### Is there any integrable function in two variables such that it is integrable for any fixed $y$ but the iterated integral does not exist?

It can be proven that Suppose $f:[a,b]\times[c,d]\to\mathbf R$ be bounded and integrable. For any fixed $y$, suppose $$A(y)=\int_a^bf(x,y)\mathrm dx$$ exists. If $\int_c^dA(y)\mathrm dy$ exists, then ...
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### Finding $P(X+Y<1)$ given $f(x,y) = 12xy^3$ when $0 < x < y < 1$ [closed]

So none of the student-assistants are able to figure this one out, but maybe you will. $$f(x,y) = 12xy^3\quad \text{for}\quad 0 < x < y < 1$$ If you draw out the figures you get that in this ...
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### nature of a triple integrale

I would like to show that $$I:=\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{sin^{6}(x_{1})}{x_{1}^{6}}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{2}dx_{1}dx_{2}dx_{2}<\infty$$ However when I try ...
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### How to change constant bounds in double integral into polar.

for example consider: $$I:=\int_{a}^{b}\int_{c}^{d}f(x,y)\,dx\,dy$$ Now let’s say I want to convert to polar. How would my bounds change? The only examples I’ve seen and could find, have been when the ...
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### How to calculate $\int_{x^2+y^2+z^2\leq R^2}\frac{dxdydz}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}}$, where $a^2+b^2+c^2>R^2$?

Let$a^2+b^2+c^2>R^2$, calculate $\int_{x^2+y^2+z^2\leq R^2}\frac{dxdydz}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}}$. Let $(x,y,z)=(a+r\sin\phi\cos \theta,b+r\sin\phi\sin \theta,z=c+r\cos\phi)$, but I don’t ...
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### Is the integral $\int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \frac{e^{i a x}}{\sqrt{x^2+y^2}}$ convergent?

Is the integral $$I = \int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \frac{e^{i a x}}{\sqrt{x^2+y^2}}$$ convergent for real $a$? I have an idea to calculate it but I am not sure if it is correct: ...
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### Find surface area of cone cut off by a cylinder

Find the surface area of cone ${x^2 + y^2 = z^2}$ cut off by surface of cylinder ${x^2 + y^2 = a^2}$ above the $xy$ plane. My approach: I considered projection of the area on $xy$ plane cut off by ...
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### Show that integral $\int_{0}^{2a}\int_{\sqrt{2ax-x^2}}^{\sqrt{4ax-x^2}}\left(1+\frac{y^2}{x^2}\right)\ dy\ dx=\left(\pi+\frac{8}{3}\right)a^2$

Show that the integral $$\int_{0}^{2a}\int_{\sqrt{2ax-x^2}}^{\sqrt{4ax-x^2}}\left(1+\frac{y^2}{x^2}\right)\ dy\ dx=\left(\pi+\frac{8}{3}\right)a^2$$ by changing the coordinates $x,y$ to $r$,$\theta$ ...
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### Evaluate the integral $\int_{0}^{2a}\int_{\sqrt{2ax-x^2}}^{\sqrt{4ax-x^2}}(1+\frac{y^2}{x^2})\ dy\ dx$ by changing the coordinates to r,$\theta$

where, $x=r\cos^2\theta , y=r\sin\theta \cos\theta$ I drew the region on the $xy$-plane over which the integral is performed After the transformation to the $R\Theta$-plane the region changes. To me ...
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### How to compute the double integral $\iint_{D}\frac{{\rm d}x \, {\rm d} y}{y-2}$?

Given the domain $$D := \left\{ (x,y) \in {\Bbb R}^2 : x^2 + y^2 \leq 4 , y\geq 0 ,y^2\geq 4(1+x) \right\}$$ calculate the following double integral $$\iint_{D}\frac{{\rm d}x \, {\rm d} y}{y-2}$$ I ...