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Questions tagged [multiple-integral]

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

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What values of $\rho$ can give the answer? $V=\int_0^{2π}\int_0^{π/2}\int_{?}^{?}\rho^2\sin\phi d\rho d\phi d\theta$

Find the Volume of the region bounded above by sphere $x^2 + y^2 + z^2 = 2a^2$, and below by the paraboloid $az = x^2 + y^2$? $$V=\int_0^{2π}\int_0^{π/2}\int_{?}^{?}\rho^2\sin\phi d\rho d\phi d\theta$...
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1answer
35 views

Triple Integral in spherical coordinate

$\displaystyle\iiint_R (x^2+y^2+z^2)^{-2}\,dx\,dy\,dz$ where $R$ is in the region in the first octant outside the sphere $x^2+y^2+z^2 = 1$; Hi guys, I don't quite get which region is this, is it that ...
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0answers
33 views

How can I calculate Fresnel integral via double improper integral?

So, there are many ways to calculate the integral $\int_{-\infty}^{+\infty} \sin(x^2)\,\mathrm{d}x$. For example, you can use differentiation under the integral sign or you can use complex numbers. My ...
3
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1answer
51 views

How should I solve this triple integral?

Hi everybody I have a triple integral I can't solve: $$\iiint \sqrt {x^2+y^2+z^2} \,dx \,dy \,dz $$ Which the region is between $z=\sqrt {x^2+y^2}$ and $z=4$ . The question says after using the ...
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35 views

Finding volume of solid under $z = \sqrt{1-x^2-y^2}$ above the region bounded by $x^2 + y^2-y=0$

Find the volume of the solid that is under the hemisphere $z=\sqrt{1-x^2-y^2}$ above the region bounded by the graph of the circle $x^2 + y^2-y=0$. I solved this problem using limits of integration $...
3
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1answer
45 views

Integration with Dirac delta function of two-argument function

I would like to solve that problem: $$ \int d^2 \mathbf{k} \, \delta(f(k,\phi)) = \int_{k_1}^{k_2} dk \, k \int_0^{2\pi} d\phi \, \delta(f(k,\phi)) \,, $$ where $f(k,\phi) = a - bk^2 - ck^3 |\sin(2\...
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1answer
35 views

Surface area of a sphere over a disc

What's the surface area of the sphere $x^2 + y^2 + z^2 = 1$ over the disc $(x-1/2)^2 + y^2 \le 1/4$ ? I've tried something, but I don't think it's right, as it's not a "nice answer" So here is what ...
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0answers
13 views

Manipulating change of variables formula for multiple integrals with roots

The change of variables for multiple integral can be formulated like: $\int_D(f \circ g) (x) |det~J_g(x)|dx = \int_{D'}fdx'$ for a continous $f: D'\rightarrow X$ and two-times continous, injective $...
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Changing order of $\int_{-23/4}^4\int_0^{4-y}\int_0^{\sqrt{4y+23}} f(x,y,z) \,dx\,dz\,dy$ to $\iiint f(x,y,z) \, dy\,dz\,dx$

The sketch of the region is: Based on it I get: $$\int_0^{\sqrt{23}}\int_0^{4-y}\int_{\frac{x^2-23}{4}}^{4} f(x,y,z)\, dy\,dz\,dx$$ But this arrangement makes me drag $y$ so I must be missing ...
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2answers
28 views

Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$

Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$ Not sure how to proceed
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Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$

I am trying to evaluate the following integral $$\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_{n}, \hspace{0.5cm} 0<...
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1answer
21 views

Upper bound for an iterated integral

The integral taken over R, the area bounded by curves y = 0, x = 1, and y = x. The function we are integrating is f(x,y) = $sin(x^{1000}y)$. So the integral is $\int_0^1 \int_0^x sin(x^{1000}y)dydx$ ...
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1answer
26 views

Multiple integration with constraints on variables

I have a function $(x_1, x_2)\mapsto g(x_1, x_2)$ where $x_1$ and $x_2$ are both 3D vectors. I would like to integrate function $g$ over the whole space but with some constraints on $x_1$ and $x_2$ ...
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Compute $\lim\limits_{r\to +\infty} e^{-r}\int_{B(0,r)} e^{|x|+|y|} \,{\rm d}x \,{\rm d}y$

The integrand function is continuous over a compact set so the integral exists finited. To compute it I use polar coordinate? \begin{align} \int f dx dy &= \int_{0}^{\pi/4} d\theta \int_{0}^{...
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1answer
23 views

compute the volume of the solid E

The points (x,y,z) of this solid projected on the plane z=0 are in the parallelogram (2,1),(6,-1),(7,0),(3,2) and $\le z \le e^{x+2y}$. $$\int_{2}^{7}dx \int_{-0.5x+2}^{-0.5x+3.5}dy \int_{0}^{e^{x+2y}...
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1answer
29 views

Differentiating under the integral sign (multiple integrals)

$\newcommand{\bexp}[1]{\exp{\left(#1\right)}}$ I am trying to derive the pdf of a certain random variable $U$ and I have something that looks like this: $$f_{U}(u) = -\frac{d}{du}\int_{x=u}^{\infty}\...
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2answers
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What is the formal definition of “ Express the double integral in polar coordinate ”?

I was reading Tom Apostol calculus volume 2 and came across, the change of variable for double integral (page 392), the exercise followed by the sections (section 11.28), the very first question of ...
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1answer
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Triple integrals in spherical coordinates, volume of octant

So, the question is : $S$ is the part of the sphere $ρ=a$ cut by the planes $\theta=0$ and $\theta=\frac{\pi}{6}$ in the first octant. Find the volume of $S$. I am taking the integration limits as $0≤...
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1answer
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How to determine the interval of $\varphi$ for integration?

I have $\int_{D_2}(x^3+xy^2+y) d(x,y)$ with $D_2 := \{(x,y)^T\in{\mathbb R^2} : 4 \leq x^2+y^2 \leq 9, x \geq 0, y \geq 0\}$. Obviously, I want to transform this into polar coordinates and determine ...
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1answer
22 views

Multiple integrals in polar coordinates

So, I was practicing some problems and found this: $$\int_0^2 \int_0^{\sqrt{4-x^2}} e^{x^2+y^2} dydx.$$ While converting from Cartesian to polar coordinates, $\theta$ limits will be from $0$ to $\pi ...
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0answers
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Is there no similar proof to change of variables for double integrals as there is for a single integral?

The single variable change of variable theorem states that, for any function $\varphi$ with integrable derivative, and a continuous function $f$: $$\int_{\varphi(a)}^{\varphi(b)}f(x)dx=\int_a^bf(\...
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3answers
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Looking for where I went wrong: Finding the volume of a solid that lies within both a cylinder and sphere

I'm currently working on this question: Find the volume of the solid that lies within both the cylinder $x^2+y^2=1$ and the sphere $x^2+y^2+z^2=4$. I decided to use polar coordinates so that the ...
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2answers
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Understanding the textbook: Expressing a triple integral in a different order

I was going over my textbook and have trouble understanding this process: Why does the innermost integral have the bounds $\sqrt{y}$ and $1$? The shape on the $xz$ plane looks the same as the one on ...
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1answer
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How to find volume of the region $\{(x,y,z)|\,0 \le (x-1)^2+y^2 \le z(1-z)\}$?

I need to find the volume of the region : $$R=\{(x,y,z)| \space 0 \le (x-1)^2+y^2 \le z(1-z)\}$$ I don't understand the region. Is it the volume outside the cylinder and inside a sphere? Can you ...
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0answers
36 views

An upper bound for a multiple integral

Suppose $0 < a < b < \infty$. I am trying to find a nontrivial upper bound for: $$ \int^\infty_0 \int^\infty_0 \frac{dx \, dy}{xy + \frac{e^{a(x+y)}}{b^2}}. \tag{1} $$ Now, I know that $$ ...
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0answers
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volume calculation

I have to solve the following exercise: Exercise. Determine the volume of the set $\Omega\subseteq \mathbb{R}^3$ $$\Omega=\bigg\{(x,y,z)\in\mathbb{R}^3\;\bigg|\;x^2+y^2+z^2\le R^2,x^2+y^2\le 1\bigg\...
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0answers
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In search of a trick for a multiple integral with difficult integration limits

I'm trying to make a small model for the expected life-time of some molecules (I'll edit the question and add info if someone wants to know the context) and I reached the following multiple integral: $...
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1answer
56 views

Compute integral in unbounded area

Compute the following integral: $$\iint_D \frac { |x-y| } { (x^2 + y^2 + 1)^2 } \, dx dy , $$ where $D = \{ (x, y) \in \mathbb R^2 \mid x \geq 0 ,\, y \geq 0 \}$. So is there an easy way to ...
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0answers
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Volume bounded by the surface $x=-1,\;x=1,\;y=-1,\; y=1, \; z=2,\; y^2+z^2=2$

Volume bounded by the surface $x=-1,\;x=1,\;y=-1,\; y=1, \; z=2,\; y^2+z^2=2$ in $R^3$ The region $z=2 $ and $y^2+z^2=2$ is confusing me. I know this is a cylinder with axis parallel to $x-axis$ and ...
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1answer
39 views

$\int_{c}(xy-y^2)dx-x^3dy$ over the region enclosed by $y=x^2, y=x+2, x=0$

$\int_{c}(xy-y^2)dx-x^3dy$ over the region enclosed by $y=x^2, y=x+2, x=0$ I'd like to use the Green's theorem to solve this, but I am not sure which region I need to take? Am I missing something or ...
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23 views

Relation between two functions defined by a line integral

Let $A$ and $B$ two subsets of ${{\mathbb R}^2}$ such that $$\eqalign{ & A = \left\{ {(t,x) \in {{(0,1)}^2},x - t \in (0,1)} \right\} \cr & B = \left\{ {(t,x) \in {{(0,1)}^2},x + t \in (...
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2answers
64 views

Sketch and find the volume of the solid in the first octant bounded by the coordinate planes, plane x+y=4 and surface z=root(4-x)

I understand that this can be done with triple integrals, but my class has yet to be taught those and we will be assessed on our ability to perform a question similar to this one with the principles ...
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2answers
56 views

Find $\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$

$$I:=\int_{0}^{1} \int_{x}^{1}y^4e^{xy^2}dy dx$$ Here the region of integration is the triangle with vertices $(0,0),(0,1)$ and $(1,1)$ and given as a type-1 region. We can convert it into a type-2 ...
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1answer
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Simplification of an expression of a double integral

Is the following expression able to be simplified? $$I = \int_0^{x/2} \int_0^s f(s,r)\,dr\,ds + \int_{x/2}^x \int_{2s - x}^s f(s,r)\,dr\,ds . $$ Here $f(s,r)=u(r,2s+x-r)$ so that we could also write $...
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0answers
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How to find the work integral?

The question is, Let $P_{1}$ and $P_{2}$ be the points at distances $s_{1}$ and $s_{2}$ from origin. Show that the work done by gravitational force $$\vec{F}=-\frac{GMm (x\hat{i}+y\hat{j}+z\hat{k})}{(...
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1answer
55 views

Triple integral $\iiint x^2 \,dx\,dy\,dz.$

Here's the question $$\iiint x^2 \,dx\,dy\,dz.$$ I am asked to evaluate this integral over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2 \leq y^2+z^2 \leq 4\right \}.$$ I've shown that : $ 2 ...
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0answers
33 views

Double integral - how to set the upper and lower bounds

Need some help with this double integral, would like to solve it using polar coordinates, but don't know how to set the upper and lower bounds when there is also absolute value. I = $\iint_D |((x+y)...
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1answer
44 views

Why does the order of integration matter for the function $f(x,y)=y$?

Consider the integrals: $$ I=\int_{0}^{\sqrt{2}}\int_{y^2}^{2}y\ dxdy \\ I'=\int_{y^2}^{2}\int_{0}^{\sqrt{2}}y\ dydx $$ From what I understand, the order of integration does not matter. However, as ...
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0answers
10 views

Probability density, bivariate

Hi i have a problem where i am really unsure if my solution is correct. I tried using Mathjax but it did not work out.....that is why i am including a picture of my solution. The problem: $f(x,y)=\...
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1answer
71 views

The double integral $\int_{1}^{2}\int_{x}^{2x}f(x,y)dxdy$ under the tranformation $x=u-uv $ and $y=uv$ is__________________

The double integral $\int_{1}^{2}\int_{x}^{2x}f(x,y)dxdy$ under the tranformation $x=u-uv $ and $y=uv$ is__________________ I have calculated jacobian as u but I am not able to find out the limits of ...
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0answers
46 views

Integration by Parts Within Multiple Integral

In the innermost iteration of a triple integral problem, I've applied two instances of integration by parts, and ended up with$$\int_0^{\pi/6}\int_0^{\pi}x\int_0^1ysin(yz)\ dy\ dx\ dz\ = \int_0^{\pi/6}...
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0answers
25 views

Integrating in a spherical region using n-dimensional spherical coordinates

I want to evaluate an integral of the form $\int_{D}f(x_1,x_2,...x_n)dx_1dx_2...dx_n$ where $D$ is the $n$-dimensional spherical region lying between spheres of radii $r$ and $R$, $r<R$. I want ...
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2answers
80 views

How do I evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-(3x^2+2 \sqrt 2 xy+3y^2)} \mathrm dx\,\mathrm dy$?

Evaluate $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp\left(-3x^2-2 \sqrt 2 xy - 3y^2\right) \, \mathrm dx\,\mathrm dy$$ I first evaluate $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \...
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1answer
37 views

Help evaluating this surface integral, how to evaluate $dS$ in this?

Evaluating this surface integral $\int_{S}(a^2x^2+b^2y^2+c^2z^2)^{1/2}dS$ over the ellipsoid $S:ax^2+by^2+cz^2=1$ I am well aware of evaluating surface integral over any given sphere $x^2+y^2+z^2=a^...
0
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1answer
28 views

For which $\alpha\in\mathbb{R}$ the integral $\int_{\mathbb{R}^{2}}\frac{dxdy}{\left(1+x^{2}+xy+y^{2}\right)^{\alpha}}$ converges/diverges?

Im looking for which $\alpha\in\mathbb{R}$ the integral $\int_{\mathbb{R}^{2}}\frac{dxdy}{\left(1+x^{2}+xy+y^{2}\right)^{\alpha}}$ converges/diverges. What I was looking for is an appropriate change ...
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0answers
45 views

What is the volume enclosed by a plane $y = 23$ and a torus of radius $27$ and inner radius $4$

I want to find the volume on the right of the plane (the smaller volume). I think that the torus can be thought of as a volume of revolution of a circle about the z-axis, and that the bounds for ...
0
votes
1answer
55 views

How to write the limits of triple integral $\iiint f(x,y,z) dz dy dx$ over the annulus?

How to write the limits of triple integral $\iiint f(x,y,z) dz dy dx$ over an annulus which lies between the circle of radii $r$ and $R$, $r<R$? I am confused. I don't want to change into polar ...
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1answer
39 views

How to find the volume of ellipsoid using substitution in triple integral?

The equation of ellipsoid is $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ The hint for substitution is given as $x=au$, $y=bv$ and $z=cv$. I know that the projection of the ellipsoid on $xy-$...
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1answer
29 views

How to find the spherical coordinates limits calculate the volume of the solid region?

The solid bounded below by the hemisphere $\rho=1,$ $ 0\leq z$ and above by the cardoid of revolution $\rho=1+\cos\phi$ I am new to the triple integral in spherical coordinates. I know that limits of ...
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0answers
44 views

For what value of “c” volume of ellipsoid equal to $8\pi$?

The equation of ellipsoid is $$x^2+\bigg(\frac{y}{2}\bigg)^2+\bigg(\frac{z}{c}\bigg)^2=1$$ I have taking the limits of integration $$\int_{0}^{1}\int_{-2}^{2}\int_{0}^{c\sqrt{1-x^2-\frac{y^2}{4}}}...