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 $az = x^2 + y^2$?
$$V=\int_0^{2π}\int_0^{π/2}\int_{?}^{?}\rho^2\sin\phi d\rho d\phi d\theta$...
0
votes
1answer
35 views
Triple Integral in spherical coordinate
$\displaystyle\iiint_R (x^2+y^2+z^2)^{-2}\,dx\,dy\,dz$ where $R$ is in the region in the first octant outside the sphere $x^2+y^2+z^2 = 1$;
Hi guys, I don't quite get which region is this, is it that ...
0
votes
0answers
33 views
How can I calculate Fresnel integral via double improper integral?
So, there are many ways to calculate the integral $\int_{-\infty}^{+\infty} \sin(x^2)\,\mathrm{d}x$. For example, you can use differentiation under the integral sign or you can use complex numbers. My ...
3
votes
1answer
51 views
How should I solve this triple integral?
Hi everybody I have a triple integral I can't solve:
$$\iiint \sqrt {x^2+y^2+z^2} \,dx \,dy \,dz $$
Which the region is between $z=\sqrt {x^2+y^2}$ and $z=4$ .
The question says after using the ...
0
votes
0answers
35 views
Finding volume of solid under $z = \sqrt{1-x^2-y^2}$ above the region bounded by $x^2 + y^2-y=0$
Find the volume of the solid that is under the hemisphere $z=\sqrt{1-x^2-y^2}$ above the region bounded by the graph of the circle $x^2 + y^2-y=0$.
I solved this problem using limits of integration $...
3
votes
1answer
45 views
Integration with Dirac delta function of two-argument function
I would like to solve that problem:
$$
\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))
\,,
$$
where $f(k,\phi) = a - bk^2 - ck^3 |\sin(2\...
0
votes
1answer
35 views
Surface area of a sphere over a disc
What's the surface area of the sphere $x^2 + y^2 + z^2 = 1$ over the disc $(x-1/2)^2 + y^2 \le 1/4$ ?
I've tried something, but I don't think it's right, as it's not a "nice answer"
So here is what ...
0
votes
0answers
13 views
Manipulating change of variables formula for multiple integrals with roots
The change of variables for multiple integral can be formulated like:
$\int_D(f \circ g) (x) |det~J_g(x)|dx = \int_{D'}fdx'$
for a continous $f: D'\rightarrow X$ and two-times continous, injective $...
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1answer
22 views
Changing order of $\int_{-23/4}^4\int_0^{4-y}\int_0^{\sqrt{4y+23}} f(x,y,z) \,dx\,dz\,dy$ to $\iiint f(x,y,z) \, dy\,dz\,dx$
The sketch of the region is:
Based on it I get:
$$\int_0^{\sqrt{23}}\int_0^{4-y}\int_{\frac{x^2-23}{4}}^{4} f(x,y,z)\, dy\,dz\,dx$$
But this arrangement makes me drag $y$ so I must be missing ...
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votes
2answers
28 views
Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$
Find the volume of the cylinder $x^2+y^2\leq 2$ bounded by the xy plane and $z=x^2+y^2$
Not sure how to proceed
3
votes
0answers
80 views
Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$
I am trying to evaluate the following integral
$$\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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votes
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 $\int_0^1 \int_0^x sin(x^{1000}y)dydx$
...
1
vote
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 $x_2$ ...
-2
votes
0answers
32 views
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}^{...
1
vote
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 $\le z \le e^{x+2y}$.
$$\int_{2}^{7}dx \int_{-0.5x+2}^{-0.5x+3.5}dy \int_{0}^{e^{x+2y}...
1
vote
1answer
29 views
Differentiating under the integral sign (multiple integrals)
$\newcommand{\bexp}[1]{\exp{\left(#1\right)}}$
I am trying to derive the pdf of a certain random variable $U$ and I have something that looks like this:
$$f_{U}(u) = -\frac{d}{du}\int_{x=u}^{\infty}\...
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votes
2answers
29 views
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 ...
0
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1answer
35 views
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≤...
0
votes
1answer
23 views
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 ...
0
votes
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 ...
1
vote
0answers
37 views
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 $f$:
$$\int_{\varphi(a)}^{\varphi(b)}f(x)dx=\int_a^bf(\...
11
votes
3answers
1k views
Looking for where I went wrong: Finding the volume of a solid that lies within both a cylinder and sphere
I'm currently working on this question:
Find the volume of the solid that lies within both the cylinder $x^2+y^2=1$ and the sphere $x^2+y^2+z^2=4$.
I decided to use polar coordinates so that the ...
8
votes
2answers
72 views
Understanding the textbook: Expressing a triple integral in a different order
I was going over my textbook and have trouble understanding this process:
Why does the innermost integral have the bounds $\sqrt{y}$ and $1$? The shape on the $xz$ plane looks the same as the one on ...
0
votes
1answer
31 views
How to find volume of the region $\{(x,y,z)|\,0 \le (x-1)^2+y^2 \le z(1-z)\}$?
I need to find the volume of the region :
$$R=\{(x,y,z)| \space 0 \le (x-1)^2+y^2 \le z(1-z)\}$$
I don't understand the region. Is it the volume outside the cylinder and inside a sphere? Can you ...
1
vote
0answers
36 views
An upper bound for a multiple integral
Suppose $0 < a < b < \infty$. I am trying to find a nontrivial upper bound for:
$$
\int^\infty_0 \int^\infty_0 \frac{dx \, dy}{xy + \frac{e^{a(x+y)}}{b^2}}. \tag{1}
$$
Now, I know that
$$
...
1
vote
0answers
28 views
volume calculation
I have to solve the following exercise:
Exercise. Determine the volume of the set $\Omega\subseteq \mathbb{R}^3$ $$\Omega=\bigg\{(x,y,z)\in\mathbb{R}^3\;\bigg|\;x^2+y^2+z^2\le R^2,x^2+y^2\le 1\bigg\...
1
vote
0answers
20 views
In search of a trick for a multiple integral with difficult integration limits
I'm trying to make a small model for the expected life-time of some molecules (I'll edit the question and add info if someone wants to know the context) and I reached the following multiple integral:
$...
0
votes
1answer
56 views
Compute integral in unbounded area
Compute the following integral:
$$\iint_D \frac { |x-y| } { (x^2 + y^2 + 1)^2 } \, dx dy , $$
where $D = \{ (x, y) \in \mathbb R^2 \mid x \geq 0 ,\, y \geq 0 \}$.
So is there an easy way to ...
0
votes
0answers
14 views
Volume bounded by the surface $x=-1,\;x=1,\;y=-1,\; y=1, \; z=2,\; y^2+z^2=2$
Volume bounded by the surface $x=-1,\;x=1,\;y=-1,\; y=1, \; z=2,\;
y^2+z^2=2$ in $R^3$
The region $z=2 $ and $y^2+z^2=2$ is confusing me. I know this is a cylinder with axis parallel to $x-axis$ and ...
0
votes
1answer
39 views
$\int_{c}(xy-y^2)dx-x^3dy$ over the region enclosed by $y=x^2, y=x+2, x=0$
$\int_{c}(xy-y^2)dx-x^3dy$ over the region enclosed by $y=x^2, y=x+2, x=0$
I'd like to use the Green's theorem to solve this, but I am not sure which region I need to take? Am I missing something or ...
0
votes
0answers
23 views
Relation between two functions defined by a line integral
Let $A$ and $B$ two subsets of ${{\mathbb R}^2}$ such that
$$\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 (...
-1
votes
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 ...
3
votes
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 ...
0
votes
1answer
31 views
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
$...
0
votes
0answers
27 views
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 $$\vec{F}=-\frac{GMm (x\hat{i}+y\hat{j}+z\hat{k})}{(...
0
votes
1answer
55 views
Triple integral $\iiint x^2 \,dx\,dy\,dz.$
Here's the question
$$\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 \}.$$
I've shown that :
$ 2 ...
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votes
0answers
33 views
Double integral - how to set the upper and lower bounds
Need some help with this double integral, would like to solve it using polar coordinates, but don't know how to set the upper and lower bounds when there is also absolute value.
I = $\iint_D |((x+y)...
1
vote
1answer
44 views
Why does the order of integration matter for the function $f(x,y)=y$?
Consider the integrals:
$$
I=\int_{0}^{\sqrt{2}}\int_{y^2}^{2}y\ dxdy \\
I'=\int_{y^2}^{2}\int_{0}^{\sqrt{2}}y\ dydx
$$
From what I understand, the order of integration does not matter. However, as ...
0
votes
0answers
10 views
Probability density, bivariate
Hi i have a problem where i am really unsure if my solution is correct.
I tried using Mathjax but it did not work out.....that is why i am including a picture of my solution.
The problem:
$f(x,y)=\...
1
vote
1answer
71 views
The double integral $\int_{1}^{2}\int_{x}^{2x}f(x,y)dxdy$ under the tranformation $x=u-uv $ and $y=uv$ is__________________
The double integral $\int_{1}^{2}\int_{x}^{2x}f(x,y)dxdy$ under the tranformation $x=u-uv $ and $y=uv$ is__________________
I have calculated jacobian as u but I am not able to find out the limits of ...
0
votes
0answers
46 views
Integration by Parts Within Multiple Integral
In the innermost iteration of a triple integral problem, I've applied two instances of integration by parts, and ended up with$$\int_0^{\pi/6}\int_0^{\pi}x\int_0^1ysin(yz)\ dy\ dx\ dz\ = \int_0^{\pi/6}...
0
votes
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 ...
1
vote
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
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \...
0
votes
1answer
37 views
Help evaluating this surface integral, how to evaluate $dS$ in this?
Evaluating this surface integral
$\int_{S}(a^2x^2+b^2y^2+c^2z^2)^{1/2}dS$ over the ellipsoid $S:ax^2+by^2+cz^2=1$
I am well aware of evaluating surface integral over any given sphere $x^2+y^2+z^2=a^...
0
votes
1answer
28 views
For which $\alpha\in\mathbb{R}$ the integral $\int_{\mathbb{R}^{2}}\frac{dxdy}{\left(1+x^{2}+xy+y^{2}\right)^{\alpha}}$ converges/diverges?
Im looking for which $\alpha\in\mathbb{R}$ the integral $\int_{\mathbb{R}^{2}}\frac{dxdy}{\left(1+x^{2}+xy+y^{2}\right)^{\alpha}}$
converges/diverges.
What I was looking for is an appropriate change ...
2
votes
0answers
45 views
What is the volume enclosed by a plane $y = 23$ and a torus of radius $27$ and inner radius $4$
I want to find the volume on the right of the plane (the smaller volume).
I think that the torus can be thought of as a volume of revolution of a circle about the z-axis, and that the bounds for ...
0
votes
1answer
55 views
How to write the limits of triple integral $\iiint f(x,y,z) dz dy dx$ over the annulus?
How to write the limits of triple integral $\iiint f(x,y,z) dz dy dx$ over an annulus which lies between the circle of radii $r$ and $R$, $r<R$? I am confused. I don't want to change into polar ...
0
votes
1answer
39 views
How to find the volume of ellipsoid using substitution in triple integral?
The equation of ellipsoid is
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$
The hint for substitution is given as
$x=au$, $y=bv$ and $z=cv$.
I know that the projection of the ellipsoid on $xy-$...
0
votes
1answer
29 views
How to find the spherical coordinates limits calculate the volume of the solid region?
The solid bounded below by the hemisphere $\rho=1,$ $ 0\leq z$ and above by the cardoid of revolution $\rho=1+\cos\phi$
I am new to the triple integral in spherical coordinates.
I know that limits of ...
0
votes
0answers
44 views
For what value of “c” volume of ellipsoid equal to $8\pi$?
The equation of ellipsoid is
$$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}}}...
