Questions tagged [multiple-integral]

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

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2answers
31 views

How to find limits of an integral in spherical and cylindrical coordinates if you transform it from cartesian coordinates

If you have a volume integral in Cartesian coordinates with given limits of x,y and z and you want to transfer it to another coordinate system like spherical and cylindrical coordinates. I can easily ...
2
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1answer
25 views

Triple Integral in Cylindrical Coordinates using x axis instead of z axis

Today in my Calculus class my teacher made an example using change of variables using this problem: Find the volume of the solid inside the cylinder $y^{2} + z^{2} = 2y$ bounded by $x=0$ and the ...
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0answers
26 views

Line integral $\int\limits_L \left(3y+e^{cos(x)}\right)\ dx + \left(7x-\sqrt[3]{y^4+10}\right)\ dy$ for $x^2+y^2=3^2$

I have attempted to calculate the line integral $$\int\limits_L \left(3y+e^{\cos(x)}\right)\ dx + \left(7x-\sqrt[3]{y^4+10}\right)\mathrm{d}y$$ where $L$ is the circle $x^2+y^2=3^2$. I decided to use ...
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0answers
35 views

Divergence theorem given $\overset{\rightharpoonup} F = x \hat{i} + y \hat{j} +z \hat{k}$ and the volume

I am trying to calculate $\iint\limits_S \overset{\rightharpoonup} F \cdot \overset{\rightharpoonup} n\ dS$ using the divergence theorem. It is given that $\overset{\rightharpoonup} F = x \hat{i} + y \...
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2answers
49 views

How do I evaluate $\int_{0}^{\sqrt{2}}\int_{x}^{\sqrt{4-x^2}}{\sqrt{x^2+y^2}} \, dy \, dx$?

I'm having troubles evaluating this double integral. Can somebody help me? I've gone to the part that I need to use trigonometric substitution, but performing the said sub, I think I'm kind of unsure ...
2
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3answers
45 views

Calculate the volume of the region in the first octant bound by $y=0$, $y=x$, $x^2+y^2+z^2=4$

I am trying to calculate the volume of the region in the first octant bound by the surfaces $$y=0,\hspace{1em}y=x,\hspace{1em}x^2+y^2+z^2=4$$ I have found that $x,\ y,$ and $z$ can have a lower bound ...
1
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1answer
118 views

Interchanging expectations of log likelihood

I see in papers (here in eq. 3 or here on page 4, for example) that it can be done like this using Fubini $$\mathbb{E}_x\mathbb{E}_\theta\log f(x|\theta)=\mathbb{E}_\theta\mathbb{E}_x\log f(x|\theta)$$...
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1answer
58 views

Calculate $\iint_D \frac{\sqrt{x^2 + y^2}}{1+x^2+y^2} dx\,dy$

I am attempting to solve the integral: $$\iint_D \frac{\sqrt{x^2 + y^2}}{1+x^2+y^2} dx\,dy\,,$$ where $D$ is bounded by $1<x^2+y^2<9$ and within the sector bounded by lines $$-\frac{y\sqrt{2}}{2}...
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1answer
36 views

Variable substitution in double integral

Let $(a,b)\in (0,1)$ and let $T>0$. Consider the following integral: $$ \iint_{\Gamma }f(x)g(x+t)\,dt\,dx, $$ where $ \Gamma =\left\{ (t,x)\in (0,T)\times (0,1):t+x\in (0,1)\right\} . $ Let $s=x+t$...
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0answers
33 views

Calculate $\iint_A\frac{1}{1+x^2+y^2}dxdy$ with $A=\{(x,y)\in\mathbb{R^2}:x^2+y^2\le1, 0\le x\}$

$A=\{(x,y)\in\mathbb{R^2}:x^2+y^2\le1, 0\le x\}$ Calculate $\iint_A\frac{1}{1+x^2+y^2}dxdy$ I'm doing that problem from an exam, and the solution is the following: Use polar coordinates, $x=r\cos(\...
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2answers
200 views

Close-form for triple integral $ \int_0^c \int_0^b \int_0^a \sqrt{x^2+y^2+z^2} dx dy dz$

I am able to work out the double integral $$\int_0^b \int_0^a \sqrt{x^2+y^2} dx dy $$ with brute-force (i.e. integrating $x$, then $y$) to arrive at the close-form result $$\frac13ab\sqrt{a^2+b^2} +\...
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1answer
28 views

How to show that (Change of variable) $ rF'(r)=\int_{\partial D(0, r)} \frac{\partial f}{\partial x}dy-\frac{\partial f}{\partial y}dx $

Let $F(r)=\int_0^{2\pi} f(r\cos \theta, r\sin \theta) d\theta$ where $r>0$. Show that $$ rF'(r)=\int_{\partial D(0, r)} \frac{\partial f}{\partial x}dy-\frac{\partial f}{\partial y}dx $$ where $\...
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1answer
23 views

Compute the integral, $\iint_R x^2+xy^3 dA$ where R is bound $0\leq y \leq 2$ [closed]

Compute the integral, $\iint_R x^2+xy^3 dA$ where R is bound $0\leq y \leq 2$ I found this question on one of my lecutre tutorials. I just want to know if its possible to calculate this without ...
3
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2answers
101 views

Any neat way to solve the integral $\int_{-a}^a \int_{-b}^b\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}}\,dxdy$?

Straight to the point, given the integral $$\iint_Q \frac{1}{\left(x^2+y^2+z^2\right)^{3/2}}\,dxdy$$ where $Q=[-a,a]\times[-b,b]$, can you think of any neat way to solve it? At a first glance it ...
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2answers
164 views

Numerically evaluating $\iiint_{C} \frac{ {\rm d}x \, {\rm d}y \, {\rm d}z}{x^2 +y^2 + z^2}$

Numerically evaluate $$\iiint_{C} \frac{ {\rm d}x \, {\rm d}y \, {\rm d}z}{x^2 +y^2 + z^2}$$ where $$C = [-1,1] \times [-1,1] \times [-1,1] = [-1,1]^3$$ I do not expect an analytic solution. I would ...
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0answers
15 views

How would a rotation affect a rectangular domain of integration?

I want to compute the following integral $$ \int_0^1\cdots\int_0^1 (a^Tx)^2\,dx_1\cdots dx_d $$ I know that it could be computed by expanding and integrating each term, but I'm not interested in that ...
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2answers
28 views

Find the volume of $V\subset \mathbb{R^3}$ limited by a plane and a paraboloid

Let $V=\{(x,y,z)\in\mathbb{R^3}: x^2+y^2\le z, z\le x+2\}$ Then the volume of V is: (A) Vol(V) = $\frac{75}{8}\pi$ (B) Vol(V) = $\frac{81}{32}\pi$ (C) Vol(V) = $\frac{9}{4}\pi$ (D) Vol(V) = $\frac{45}{...
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1answer
27 views

Calculating the volume of a region in $\mathbb{R}^3$

We are interested in the volume of the bounded region above $z = \sqrt{3x^2 + 3y^2}$ inside $x^2+y^2+z^2=9$. To Find the volume, we need to calculate this: $$ \iint_{x^2+y^2=3} \sqrt{9-x^2-y^2} - 3\...
2
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1answer
45 views

compute $ \iiint_Kxyz\ dxdydz$

The question is: $$ \iiint_Kxyz\ dxdydz\quad k:=\{(x,y,z):x^2+y^2+z^2\leq1, \ \ x^2+y^2\leq z^2\leq 3(x^2+y^2), \ x,y,z\geq 0\} $$ Here how i have tried to solve this: $$\iint_{x^2+y^2\leq1}\int_{\...
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1answer
23 views

Moment of Inertia of a Lamina Around the Center of Mass

I have a two-dimensional lamina in the $xy$-plane, and I need to calculate the moment of inertia around the center of mass. I know that the moment of inertia around the $x$-axis is $I_x = \int\int y^2 ...
4
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1answer
79 views

Calculate the integral $ \iint_R\sqrt{y/x}\,e^{\sqrt{xy}}\,dA$

The $R$ is the region of integration, described in the following image. To solve this integral we make a change of coordinates with $u=\sqrt{xy}$ and $v=\sqrt{\frac{y}{x}}$. Furthermore, in the ...
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2answers
35 views

Find the volume of a sphere with triple integral

Consider this figure: As shown in the figure. The center of the sphere is at $(0,0,-2)$ and the radius of the sphere is $4$. My question is how to calculate the volume of the sphere when $z\geq 0$ (...
2
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3answers
45 views

PDF of a rectangle

If I want to find the $c$ of a $PDF$ when it's given: $f_{X,Y}\left(x,y\right)=c\:\:\:\:\left(The\:area\:in\:blue\right),\:otherwise:\:0$ I try to do that: $$\int _{\frac{1}{2}}^1\:\int _{-x+\frac{3}{...
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1answer
19 views

Calculate Integral $\int_{0}^ady \int_{y}^a \sqrt{x^2-y^2}f'''(a-x)dx$

I need to show that $\int_{0}^ady \int_{y}^a \sqrt{x^2-y^2}f'''(a-x)dx=\frac{\pi}2(f(a)-f(0)-af'(0)-\frac{a^2}2f''(0))$ I started first by doing integration by parts $\int_{0}^ady \int_{y}^a \sqrt{x^2-...
1
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1answer
24 views

Triple integral with cylinder and sphere intersection domain

Let $f(x,y,z)=z$ and $$D=\{(x,y,z)\in\mathbb{R}^3:\ y\geq0,\ z\geq0,\ x^2+y^2+z^2\leq2,\ x^2+y^2\leq1\}.$$ Then $$\iiint_Df(x,y,z)dxdydz$$ is (hint: use cylindrical coordinates) (A) $\frac{3\pi}{2}\...
1
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1answer
30 views

Variable transformation in evaluating a double integral

Evaluate $$\iint_R(x+y)\,dA$$ Where $R$ is the trapezoidal region with vertices at $(0,0),(5,0),(5/2,5/2)$ and $(5/2,−5/2)$, using the transformation $x=2u+3v$ and $y=2u−3v$. Approach: The Jacobian ...
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1answer
46 views

Calculate Integral $\iiint_{K}\frac{1}{(z+1)}\,dx \,dy \,dz $ [closed]

I need to calculate this integral $$\iiint_{K}\frac{1}{(z+1)}\,dx \,dy \,dz $$ with $K=x^{2}\le y\le z\le x$ I have trouble visualizing $K$ so I am unable to find the extremities of the integrals Any ...
2
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2answers
33 views

Solve the Integral in the following set

I have the Set $D=\{ (x,y) \in\mathbb{R}: |x|\le2,|x|\le y \le \sqrt{4-x^2}\}$ and I have to calculate $\int_{D}(x^2y+xy^2) dydx$. So I cannot explain how hard it is for me to find the borders of the ...
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0answers
7 views

Sequence of Integrals converges to supremum [duplicate]

$M\subseteq \mathbb{R}^n$ is an open set with Jordan-measure $|M|\not= 0$ and $f:M\rightarrow \mathbb{R}$ is continuous, bounded and not-negative. Define following sequence: $$a_n=\left(\int_M f(x)^...
2
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1answer
100 views

How to evaluate $\int_{0}^{1}\int_{0}^{1} \sqrt{1 + 4(x^2 + y^2)}\,dx\, dy$?

I need to solve the integral $$\int_{0}^{1}\int_{0}^{1} \sqrt{1 + 4(x^2 + y^2)}\,dx\,dy$$ I am using polar coordinates here to get : $$ \int_{0} ^{\pi/4}\int_{0}^{\sec \theta} \sqrt{(1 + 4r^2)} r \,dr ...
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2answers
59 views

Changing variables double integral

I want to know if there's a way to know the best variables change for an integral calculation. For example, if we consider the integral $$ \iint_{R}\left(x+y\right)dx\,dy $$ Where $ R $ is bounded by ...
3
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2answers
74 views

Equality of triple integrals over unit sphere $ \iiint_{\text{unit ball}} x e^{ax + by + cz} dV$

I have to calculate $$ \iiint_{\text{unit ball}} x e^{ax + by + cz} \,dV,$$ where by "unit ball" I mean the region $x^2 + y^2 + z^2 \leq 1$. I know how to calculate this (rotation matrix ...
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0answers
48 views

Show the equality between two integrations

$$\int ^{\infty}_{0}\int^{\infty}_{0} e^{-t[1+v]} v^{-s}\,dvdt=\int^{\infty}_{0} \frac{v^{-s}}{1+v}\,dv$$. This shows up in one step of the proof of some property of gamma function in Stein & ...
0
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1answer
45 views

$\int_{0}^{\pi^2}\int_{x^{1/2}}^{\pi} \sin ( x/y)\, dy\,dx$ [closed]

I need to solve this integral and I don't know Please, I can't change the order of integration. I feel like it's impossible. $$\int_0^{\pi^2}\int_{x^{1/2}}^\pi\sin\frac xy\,dy\,dx$$
2
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2answers
30 views

Finding $\int_{S}^{} x^{4} \sin (x^{3}z^{5})\,dx\,dy\,dz$ where $S$ is part of a sphere

Let $S$ be the subset of the sphere $x^{2} + y^{2} + z^{2} = 1, z > 0$. Calculate the integral $$\int_{S}^{} x^{4} \sin (x^{3}z^{5})\,dx\,dy\,dz$$ So I know that this is a surface integral. I ...
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2answers
52 views

Solve $\int_{0}^{1}\int_{0}^{x}\int_{z}^{x}x\cos (y^{2})dydzdx$

I want to solve this triple integral: $$\int_{0}^{1}\int_{0}^{x}\int_{z}^{x}x\cos (y^{2})dydzdx$$ I tried to change the variables but I still couldn't make it. Is there any clue how to start?
4
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4answers
91 views

Double integral comes out different after changing $dx$ with $dy$.

$$\int_{0}^{1}\,dx \int_{0}^{1}\frac{x-y}{(x+y)^{3}}\,dy$$ My maths teacher gave this question to explain that if you change $dy$ with $dx$ the integral will have different value. My opinion is that ...
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0answers
21 views

Double Intgral using change of Varibale

Q- show that $$\iint_D \left(({1-x^2/a^2-y^2/b^2})^mF(Ax+By)\right)dxdy = {\beta(1/2,m+1)}*(\int^1_{-1} {(1-x^2)^{m+1/2} F(Kx)}dx)$$ where D is a region covered by ellipse$\frac{x^2}{a^2}+\frac{y^2}{...
2
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1answer
60 views

Is this function defined over a rational rectangle integrable?

Let $Q_0 = \{ x : x\in \mathbb{Q} , 0\leq x \leq 1 \}$ and let $f:R_Q \to \mathbb{R}$ with $R_Q = Q_0 \times Q_0$. f defined as: $$ f(x,y)= \left\{ \begin{array}{lc} \sin(x) & \...
2
votes
1answer
131 views

Multiple integration with Dirac delta

I am reading a paper and trying to solve collision integral. In Appendix A, there is an integral, $$ I^{(n)} = \int \frac{p^2_1\,dp_1 \,p^2_3\,dp_3\,p^2_4\,dp_4} {2E_12E_32E_4} 2\pi \delta (p_1 − p_3 −...
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3answers
38 views

Region of integration for double integral:

When tasked with reversing the order of integration in the following integral: $$\int^1_0\int^{y^2}_{-y}x\ \mathrm{d}x\mathrm{d}y$$ It seems sensible to sketch out our situation: we have that $-y\le x\...
3
votes
2answers
70 views

Evaluate the I?

This is question is inspired from this find the value of $$I=\int_{0}^{2\pi} \int_{0}^{\sqrt 2 a}\sqrt{ \frac{u^4}{a^2} +u^2} du dv$$ My attempt : $$I=\int_{0}^{2\pi} \int_{0}^{\sqrt 2 a}u\sqrt{ \...
0
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0answers
17 views

Trouble calculating integral on 2d probability density function

Given the following joint pdf: $$f_{X,Y}(x,y)=\begin{cases} c(1-x-y), & x,y\ge 0,x+y\le 1\\ 0, & \text{else} \end{cases}$$ I need to find $c$. I'm calculating the double integral with bounds ...
1
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1answer
32 views

Calculating $ \int_{E}(y-3) \, dx \, dy \, dz$

So I have the set $E$ as the smallest polyhedron that contains the points $(-1,0,0), (0,2,0), (0,-2,0), (2,0,0)$ and $(0,0,3)$. And I have to calculate $$ \int_{E}(y-3) \, dx \, dy \, dz$$ In all of ...
0
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1answer
32 views

Find the value of $\iint_R(f(x,y) + 2) \,dx\,dy$ over area $[0,2] \times [0,2]$

So I'm taking a calculus course and I admit I don't quite understand this topic in my course. Given that the integral of the function $f(x,y)$ over the area $[0,2] \times [0,2]$ equals $4$ , how can ...
1
vote
1answer
19 views

Change of Variable finding simple bounds

I have a very simple problem that I've solved before (and it gave me a lot of trouble then) and I somehow can not come up with the right bound again. In Michael Corral's book he uses a change of ...
0
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2answers
29 views

Volume of a cone given constraints on $z$

This question is pretty straightforward, however, it's been a couple years since I've had to solve an integral like this! Compute the volume bounded by $x^2 + y^2 \le z^2$ such that $c_1 \le z \le c_2$...
0
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1answer
40 views

Calculating triple integral for 3d body

We have a 3D body $G=\left \{ (x,y,z):x^2+y^2+z^2\leq a^2,0\leq z\leq\sqrt{x^2+y^2}\right \}$ Defining $I_0=\iiint_G \,dx\,dy\,dz$, $I_1=\iiint_G z\,dx\,dy\,dz$ For which $a\in\mathbb{N}$ we get ...
0
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0answers
13 views

Parametrization of a subset of the 3-dimensional Euclidean space

I have to compute an integral of the form \begin{equation}\int_R f(x_1, x_2, x_3)\text{ d}x_1\text{d}x_2\text{d}x_3\end{equation} where $x_i$ are scalar, $f:\mathbb{R}^3\mapsto\mathbb{R}$, and the ...
0
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1answer
33 views

Find $\oint_S (x \hat i+y\hat j+z^2 \hat k)\cdot\hat n \,dS$ where $S$ is the surface bounded by $x^2+y^2=z^2$ and the plane $z=1$.

Find $$\oint_S (x \hat i+y\hat j+z^2 \hat k)\cdot\hat n \,dS$$ where $S$ is the surface bounded by $x^2+y^2=z^2$ and the plane $z=1$. By divergence theorem the integration is $$\begin{align}\oint_S (...

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