Questions tagged [multiple-integral]
For questions regarding computation and results related to integrals in at least 2 variables.
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How to calculate the surface area of the elliptical paraboloid given by $z=x^2 + 4y^2$
Also, $z$ should lie in the closed interval of $[0,4]$.
I know the general method to this - to find the function $\langle x, y, z(x,y)\rangle$ differentiated with respect to $x$, then with respect to $...
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Let $E$ be the smaller of the two solid regions bounded by the surfaces $z=x^2+y^2$ and $x^2+y^2+z^2=6$. Evaluate $\iiint(x^2+y^2)dV$. [closed]
I tried solving and came up with something like the integral in the image given
$$\int_{\theta=0}^{2\pi} \int_{r=0}^{\sqrt{2}} \int_{z=r^2}^{\sqrt{6-r^2}} r^3 \, dz \, dr \, d\theta$$
But I cant find ...
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Applying the fundamental theorem of calculus in double integral
How would one go about calculating
$$\frac{d}{dt}\int^t_{-t}f(z,t)dt.$$
And more specifically,
$$\frac{d}{dt}\int^t_{-t}\int^t_{-t}f(x,y)dxdy$$
Assuming the necessary conditions, i got to
$$lim_{h\to ...
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Evaluating a double integral bounded by a unit disk [closed]
What would be the bounds for my double integral?enter image description here
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How do I solve the double integral given below? [closed]
I just finished my exam and no matter the methods I used I just could not solve this integral:
We are to compute the generalized double integral
$$\int{\int_S \sqrt{x^2 + y^2}e^{-(x^2+y^2)}} dxdy$$
...
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Convergence speed of the limit from sum on $\mathbb{Z}^d$ to integral on $\mathbb{R}^d$
I want to know the convergence speed of the following:
$$\lim_{\|x\|\to\infty}\sum_{z\in\mathbb{Z}^d}\frac{\|x\|^{d-4}}{\|z\|^{d-2}\|x-z\|^{d-2}}=\int_{\mathbb{R}^d}\frac{dt}{\|t\|^{d-2}\|h-t\|^{d-2}}$...
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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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1
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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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Finding the volume of the solid
I have a solid $E$ below that I would like to compute the volume. I can slice $E$ by cross-sections $E_x$ that are perpendicular to Ox. Then
$$
|E|=\int_0^1 E_x \,dx = \int_0^1 \frac{1}{2}z[e^{2x}-(-...
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Volume of the region limited by $z=1-x^2$, $y+z=1$ and $x,y,z>0$
I plot the region, but can't add a picture here... From the picture, I would say that the desired volume is
$$ V = \int _{x=0} ^1 \int _{y=0} ^\infty \int _{z=0} ^{1-x^2} 1 \ dzdydx $$
But, of course, ...
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evaluating $\iiint\limits_E x \; dV$; $E$ between $x=ay^2+az^2$ and $x=a$
$\iiint\limits_{x=ay^2+az^2}^{x=a} \!\!\!\!\! x \; dV$ (is writing integral limits like this fine?)
in my head this volume looks like an x-axis paraboloid up to the unit circle
I try to use ...
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Double integration on $D=\{(x,y)\in \mathbb{R}^2 :\sqrt{x}\leq y \leq 2\sqrt{x},\ x^2\leq y \leq 4x^2 \rbrace$. Am I right?
The problem is to integrate $x^{-3}$ on the region $D=\lbrace (x,y)\in\mathbb{R}^2 : \sqrt{x}\leq y \leq 2\sqrt{x},\ x^2\leq y \leq 4x^2 \rbrace$. I want to know if my attempt is good: Since the most '...
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Finding the domain of a triple integral lying between two surfaces
I am having trouble finding the bounds for $z$ in the domain $S$. Where $S$ lies in the first octant ($x,y,z ≥ 0$) between the two surfaces $z = x^2+y^2$ and $x^2+y^2+z^2=2$. I have drawn the domain ...
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Relationship between surface integral and fubini's theorem
I am studying about surface integral and recently have studied about fubini's theorem.
And I think there exists any relationship between 'surface integral' and 'fubini's theorem'.
As a result, I came ...
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Conditional expectation and computation of double integrals
I consider $X,Y$ two independent uniform on $[0,1]$ and $M=\min(X,Y)$.
I want to compute $\mathbb{E}(X^{2} | M]$ using the orthogonality relation that caracterizes the conditionnal expectation. For ...
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Multiple integral upper bound with nice behaviour
Consider the problem of finding a nice upper bound to the following.
$$
I:=\int_{\mathbb{R}^N} \pi^{-N/2} \frac{\prod_{i=1}^N \exp(-(x_i-a)^2)}{\sum_{i=1}^N \exp(-x_i^2)+1}dx_{1:N}
$$
I would like an ...
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Convert triple integral in cylindrical coordinates to spherical coordinates
I have this math problem right now, and any way I look at it, it just seems impossible. I don't understand.
So I am given the integral:
$$
\int_0^{2\pi}\int_0^{1}\int_0^{\sqrt{4-r^2}} r^2\,
dz\,dr\,d\...
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2
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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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Double integral and existence of limit
For each $t \in (0,1)$, the surface $P_t \in \mathbb{R}^3$ is defined by $P_t = \{(x,y,z) : (x^2 +y^2)z = 1, \, t^2\le x^2 +y^2 \le 1\}$. Let $A_t$ be the surface area of $P_t$. Then $\lim_{\{t\...
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Mass of an ellipsoid with variable density
So I'm given the problem "An object fills the ellipsoid $x^2 + y^2 + 2z^2 = 1$ and that its density is given by $r^2 \sin(\theta)$" and asked to solve for the mass.
Converting to Spherical I ...
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The moment of inertia: limits for angles.
I have this task: Calculate the moment of inertia of a homogeneous body
$G$, bounded by the surface $\{(x^2+y^2+z^2)^2=a^3z,\, a>0\}$ (image) relative to the axis of the application $OZ$. In the ...
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Double integral giving different results after changing order of integration
A double integral defined by $$\iint\frac xy dA$$ is integrated over the region of $1<x<3, x<y<2x$. Using different orders of integration produces different results as demonstrated below:
$...
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Integral inequality from AMM 1992
I would like to know the solution of the following 1992 AMM problem:
Let $f$ be a continuous non-negative function defined on the square $[0,1]^2$. Show that
$$
\int_0^1\int_0^1\int_0^1\int_0^1f(x_1,...
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2
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Why is this domain wrong?
I was trying to integrate:
$$\iiint_D x\ dx\,dy\,dz$$
where $D$ is limited by $x=4y^2+4z^2$ and $x=4$.
I have managed to find the answer by converting the domain to cylindrical coordinates, using:
$$z ...
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Evaluate double integral (find square of areas)
Given surface area of the cylinder $$x^2 + z^2 = a^2$$ cutted by the cylinder $$y^2 = a(a-x) .$$
Find the area of this surface
Solution:
First, let's visualize the problem. The cylinder $$x^2 + z^2 = ...
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Solving double integrals of sin and cosine using Fourier integral theorem
I ran into a set of four double integrals that I would like to find analytic solutions for. They are:
$$I_1=\int_0^\infty dy \int_0^\infty dx f(x) \cos(xy)\cos(yz)$$
$$I_2=\int_0^\infty dy \int_0^\...
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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 ...
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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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2
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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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Analyticity of a function in two complex variables
Let $f$ be a function defined on $\mathbb{C}^2$ given by
$$
f(s,t)=\int\limits_{-\infty}^{\infty}dk_1 \int\limits_{-\infty}^{\infty}dk_2 \int\limits_{-\infty}^{\infty}dk_3 \frac{1}{\left(\sqrt{s}-k_1\...
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How does one calculate the area of a set?
The set is $M=\{(x,y)\in\mathbb{R}^2:|x|+|y|\leq 1\}$.
Question: How do you calculate the area of $M$? More specific, how do you find the bounds of integration?
Attempt: I tried to solve the ...
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2
answers
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Finding $\lim_{c \rightarrow 0} \iint_{R} \frac{1}{(x^2+y^2)^{3/4}}\,dA$ where $R$ is unit disk with square removed
For each $0 \leq c \leq \frac{1}{\sqrt{2}}$ define the following region
$$R=\{(x,y): x^2+y^2 \leq 1\} \setminus ([-c,c] \times [-c,c])$$
Compute
$$\lim_{c \rightarrow 0} \iint_{R} \frac{1}{(x^2+y^2)^{...
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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 ...
0
votes
1
answer
116
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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 ...
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1
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How to calculate an integral with an unknown number of integration variables?
How to calculate the following integral, which has an unknown number of integration variables?
$$
\int\limits_{-\infty}^{+\infty}\cdots\int\limits_{-\infty}^{+\infty}\exp\left[-\dfrac{1}{2\theta}\sum_{...
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2
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Calculating Triple Integral using Cylindrical Coordinates
I'm given $ E $ is located in $ x^2 + y^2 = (z-1)^2 $ and between $z = 0$ and $z=2$.
I used level curves to graph this out, and as I see it is a circular cone. First, I set up my region, $$ E = \Big\{(...
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0
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Need help in detailing the proof of the existence of the test function
For every compact subset $K\subset \mathbb R^n$ and every $\epsilon>0$
$\exists$ a test function $\psi\in C_c^{\infty}(\mathbb R^n)$ such
that
(a) $0\le\psi(x)\le 1\forall x\in \mathbb R^n$
(b) $\...
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About the integral bounds of Jacobian
Given D={(x,y)|x⩾0,y⩾0,x+y⩽1}, find the value of $\iint _{D} e^{x+y} dxdy$
Method 1:
\begin{array}{l}
\iint _{D} e^{x+y} dxdy=\int _{0}^{1}\int _{0}^{1-y} e^{x+y} dxdy\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \...
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Calculate the volume of $G=\{(x,y,z) \in \Bbb R^3 : x^2+y^2+z^2 \leq 16 , 0 \leq z \leq 2 \}$
Calculate the volume of $G=\{(x,y,z) \in \Bbb R^3 : x^2+y^2+z^2 \leq 16 , 0 \leq z \leq 2 \}$
since they ask for volume we need $$\iiint_v1\,dV$$
in the solution in the book they used spherical ...
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0
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Integration over a shifted circle $\iint_D \frac{1}{\sqrt{x^2+y^2}} d y\, d x$ where $D$ is a shifted parametrised disc.
Consider the disc $D$ is centred at $(\bar{r},0)$ with radius $c$.
Illustration of $D$ :
The inner integration can be easily dealt with by using polar coordinate. However the difficulty is the ...
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3
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1
answer
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multiple integral on domain
The Question :
Calculate
$$\int_A z \ dx \ dy \ dz$$
Where $A$ is the cone : $\ A=\{0\leq z \leq 1,\ z^2\geq x^2+y^2 \}$
My try :
I first try to establish the bounds.
$z$ is from $0$ to $1$.
Once $z$ ...
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Evaluate $\iiint\limits_{\Omega}^{} \left ( x^2+y^2+z^2 \right ) dV$ and flux integral $\iint_{\partial \Omega}^{} F\cdot \overrightarrow{n} dS$
Let F be a vector field $F = \left \langle x^3,y^3,z^3 \right \rangle $ and $\Omega$ be the solid region in $R^3$ bounded by $$x^2+y^2\ge z^2,\space x^2+y^2+z^2\le 9,\space y\ge \left | x \right | .$$...
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