What is Volume integral: Definition and 50 Discussions
In mathematics (particularly multivariable calculus), a volume integral refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities.
In Greiner's Classical Electromagnetism book (page 126) he has a derivation equivalent to the following.
$$\int_V d^3r^{'} \nabla \int_V d^3r^{''}\frac {f(\bf r^{''})}{|\bf r + \bf r^{'}- \bf r^{''}|}$$
$$ \bf z = \bf r^{''} - \bf r^{'} $$
$$\int_V d^3r^{'} \nabla \int_V d^3z \frac {f(\bf z +...
(a) i sketched a quarter of a sphere centred at x=0 , y=2 , z=0
(b ) ∫ ∫ √ (4-x2 - (y-2)2) dx dy with limits 0 < x < 2 and 0 < y <4
(c ) i converted to spherical polars and took the integrand as 1/r2 . the volume element is r2sinθ drdθd∅
This leads to the triple integral of sinθ with...
Homework Statement
1) Calculate the density of states for a free particle in a three dimensional box of linear size L.
2) Show that ##\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x## provided that ##lim_{r \rightarrow \inf} [f(x)g(x)]=0##
3) Calculate the integral ##\int...
##\mathbf{M'}## is a vector field in volume ##V'## and ##P## be any point on the surface of ##V'## with position vector ##\mathbf {r}##
Now by Gauss divergence theorem:
\begin{align}
\iiint_{V'} \left[ \nabla' . \left( \dfrac{\mathbf{M'}}{\left| \mathbf{r}-\mathbf{r'} \right|}...
Homework Statement
If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##.
The Attempt at a Solution
From gauss divergence theorem we know
##\int...
Homework Statement
Find the value of the solid's volume given by the ecuation 3x+4y+2z=10 as ceiling,and the cilindric surfaces
2x^2=y
x^2=3*y
4y^2=x
y^2=3x
and the xy plane as floor.The Attempt at a Solution
I know that we have to give the ecuation this form:
∫∫z(x,y)dxdy= Volume
So, in fact...
Homework Statement
For the vector field F(r) = Ar3e-ar2rˆ+Br-3θ^ calculate the volume integral of the divergence over a sphere of radius R, centered at the origin.
Homework Equations
Volume of sphere V= ∫∫∫dV = ∫∫∫r2sinθdrdθdφ
Force F(r) = Ar3e-ar2rˆ+Br-3θ^ where ^ denote basis (unit vectors)...
Homework Statement
Calculate the electrical energy required to assemble a spherical volume of radius R and charge Q, homogeneous density ρ
the answer is (3/5)Q/R
the textbook says you have to build the volume integral one layer of sphere at a time, I'll get back to that later. I like...
Homework Statement
Calculate the volume integral of the function $$f(x,y,z)=xyz^2$$
over the tetrahedron with corners at $$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$
Homework Equations
I was able to solve it mathematically, but still can't figure out why the answer is so small.
I only understand...
I'm using the textbook Electricity and Magnetism by Purcell. In the section about continuous charge distributions I found the following formula
\mathbf{E}(x,y,z)= \frac{1}{4\pi\epsilon_0 } \int \frac{\rho(x',y',z')\boldsymbol{\hat r} dx'dy'dz'}{r^{2}} .
It's stated that (x,y,z) is fixed...
Homework Statement
Basically, this is part C of a question where in part A we had to use the RHS of the divergence theorem below to calculate the LHS, and then in part B we had to calculate the divergene of F, which came to be 0. and part C asks us how can this be? Since in part A we used the...
Homework Statement
OK, I thought once I knew what the question was asking I'd be able to do it. I was wrong!
Consider the volume V inside the cylinder x2 +y2 = 4R2 and between z = (x2 + 3y2)/R and the (x,y) plane, where x, y, z are Cartesian coordinates and R is a constant. Write down a triple...
Hi everyone.
I've been curious about a particular symbol, but I've never seen it used or mentioned in any context. I don't really have much information about its usage, so I thought I would ask around and see if anyone knew about its application.
I saw this symbol in Microsoft word.
How...
Homework Statement
Find the volume V of the region that lies inside the quarter cylinder
0 ≤ r ≤ 1, 0 ≤ θ ⇐ 1/2 π and between the planes x+y+z=4 and z=0, where (r, θ, z) are cylindrical polar coordinates.
Homework Equations
integral dV = integral r drdθdz
The Attempt at a Solution
I...
Homework Statement
Integrate $$\int_V \delta^3(\vec r)~ d\tau$$ over all of space by using V as a sphere of radius r centered at the origin, by having r go to infinity.
Homework EquationsThe Attempt at a Solution
This integral actually came up in a homework problem for my E&M class and I'm...
In 2 dimensions
given a scalar field f(x,y)
is possible to compute the line integral ##\int f ds## and area integral ##\iint f d^2A##.
In 3D, given a scalar field f(x,y,z)
is possible to compute the surface integral ##\iint f d^2S## and the volume integral too ##\iiint f d^3V##...
Hi, I have a book that makes the equality.
\vec{B}dV = (\vec{e_1}B_1 + \vec{e_2}B_2 + \vec{e_1}B_2)dx_1 dx_2 dx_3 \\[1ex]
= dx_1 \vec{e}_1(B_1 dx_2 dx_3 ) + dx_2 \vec{e}_2(B_2 dx_1 dx_3 ) + dx_3 \vec{e}_3 (B_3 dx_1 dx_2) = (\vec{B}\cdot d\vec{S}) d\vec{l}.
I'm a bit confused as to how it...
Hello.
Homework Statement
Basically I want to evaluate the integral as shown in this document:
Homework Equations
The Attempt at a Solution
The integral with the complex exponentials yields a Kronecker Delta.
My question is whether this Delta can be taken inside the integral...
Hi,
I was attempting a volume integral question out of a book. I know what the final answer is and what integral i am supposed to work out but I do not know how I am supposed to solve it. I have tried different ways such as integration by substitution and integration by parts but I do not seem...
In page 3 of this articlehttp://faculty.uml.edu/cbaird/95.657%282012%29/Helmholtz_Decomposition.pdf
I have two question:
I use ##\vec r## for ##\vec x## and ##\vec r'## for ##\vec x'## in the article.
\vec F(\vec r)=\frac {1}{4\pi}\nabla\left[\int_{v'}\nabla'\cdot\left(\frac{\vec F(\vec...
hi
http://www.calcchat.com/book/Calculus-ETF-5e/
part b they want you to rotate it about the y-axis and in part c about the line x = 3.
I don't understand this difference in writing for part b... 3^2 - (y^2)^2
And in part c they write (3-y)^2 I don't get it.
It is chapter 7 section 2...
Homework Statement
∫∫∫∇.Fdv over x2+y2+z2≤25
F= (x2+y2+z2)(xi+yj+zk)
Homework Equations
∫∫∫∇.Fdv = ∫∫ F.n dσ
n=∇g/|∇g|
The Attempt at a Solution
g(x,y,z)=x2+y2+z2-25
taking the surface integral and replacing all
(x2+y2+z2) with 25
i got
125 * ∫∫ dσ = 12500π
But...
Formulas:
Shell Method: dV = 2pi(radius) * (height) * (thickness)
Disk method: dV = pi(radius)^2 * (thickness)
Question 1 (26-3-15)
Statement
Using Shell method, find the volume generated by revolving the region bounded by the given curve about the x-axis.
x = 4y - y^2 - 3, x = 0...
Can I calculate the volume of any function rotated once about the y-axis by multiplying the definite integral of that function by 2*pi*r?
For example if we want to generate a solid 3d shape from the function -x^2+1 we multiply the integral of it, (x - x^3 / 3), by 2*pi*1. The reason r is one in...
Hi! I'm used to integrating over infinite spaces when working with QFT so far, but in an exercise I stumbled across a statement that
\int_V d^3x e^{-i \vec x \cdot (\vec p - \vec p')} = V \delta_{\vec p \vec p'}
It is clear that this is okay when p = p', but it does not seem to make sense...
Homework Statement
By making two successive simple changes of variables, evaluate:
I =\int\int\int x^{2} dxdydz
inside the volume of the ellipsoid:
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=R^{2}
Homework Equations
dxdydz=r^2 Sin(phi) dphi dtheta dr
The...
Hi
I'm working on area and volume integrals. I was wondering, when you convert to do the integral in polar, cylindrical or spherical co-ordinates, is there a standard set of limits for the theta variable in each case?
for example from 0 -pi for polar, 0-2pi for cylindrical?
If not how...
Homework Statement
The soot produced by a garbage incinerator spreads out in a circular pattern. The depth H(r) in millimeters, of the soot deposited each month at distance r kilometers from the incinerator is given by H(r) = 0.115e^(-2r)
Write the definite integral for the amount of soot...
I have the cone x^2 + y^2 <= z^2 with |z| <= 2
The vector function F = (4x, 3z, 5y)
With the divergence theorem I managed to reduce the equation to
∫∫∫ 4 dxdydz
Now the problem is finding out the limits. I know z goes from 0 to 2, but what about x and y?
Homework Statement
i have the region given as being bounded by x2+y2=4 and z=0 and z=3.
this problem asks to prove gauss divergence theorem for a given vector F
Homework Equations
The Attempt at a Solution
As for the volume integral, i had no problem. But for the surface integral, how...
find the volume inclosed by z=0 x=y and y^2+z^2=x
??
i am having trouble drawing it
i know i should look on th shadow of it on the x-y plane and integrate by it
i can project on every plane i want to and build the integral appropriatly
i jast can't imagine this shape
if i can find...
find the volume inclosed by z=0 x=y and y^2+z^2=x
i am having trouble drawing it
i know i should look on th shadow of it on the x-y plane and integrate by it
Homework Statement
[PLAIN]http://img9.imageshack.us/img9/4537/unledow.png
Homework Equations
The Attempt at a Solution
Hi, does anyone know how to find the integral that needs to be evaluated here? I can't understand how to find it from the region
edit: Oh and this is from here, not a take...
I have been thinking about the meaning of integrals and derivatives. For instance, the area of a sphere is 4 pi r^2. I can get that. The derivative of the area is 8 pi r or 4 times the circumference of the sphere. The derivative of this is just 8 pi. I can kind of understand that too. Then you...
Homework Statement
I read an e-book about the classical mechanics and didn't know how to find the lower and upper limits for a volume integral.
Homework Equations
Perhaps, it may be related to the use of similar triangle.
The Attempt at a Solution
I calculated the x value...
Homework Statement
I'm stuck on the following vector integral
Q(x)=INT[(p(y)/|x-y|)(dy)^3
For a sphere of uniform p (so it is not a function of y in this case). Where x is the position vector of a point lying outside the sphere and y is the position vector of a point lying inside the sphere...
Hi.
Can anyone tell me what the volume integral of the current density is? I find it strange, but G.D. Mahan uses it in his book on page 30. He claims that this is in fact the current. I have attached the particular page.
Homework Statement
Set up the volume integral of x^2 + y^2 + z^2 between z=sqrt(1-x^2-y^2) and z=x^2
Homework Equations
Cartesian differential volume element, spherical differential volume element
The Attempt at a Solution
In class I've been told when doing triple integrals to try...
Hi all,
evaluating \int\int\int \nabla . V d\tau over
x^{2} + y^{2} \leq 6, 0 \leq z \leq 10
where V is a vector function of just \hat{x} and \hat{y}.
Using the divergence theorem, and doing the dot product of V with the normal of the first surface,
the two partials w.r.t x and y are...
Homework Statement
"A solid cone is bounded by the surface \theta=\alpha in spherical polar coordinates and the surface z=a. Its mass density is p_0\cos(\theta). By evaluating a volume integral find the mass of the cone.
Homework Equations
The Attempt at a Solution
I can't figure...
Volume Integral! Help!
I need help with this:
Find volume of figure bounded with surface (x^2+y^2+z^2+1)^2=8*(x^2+y^2)
I tried Ostrogradsky, and spherical coordinate system with it, but I can't find boundaries...
PLEASE! HELP ME!
Homework Statement
Find the volume of the portion of the sphere x^2 + y^2 + z^2 = 4 for which y>1 (or equal to).
Homework Equations
I would like to do this using a triple integral.
The Attempt at a Solution
OK, so I tried integrating the element r^2sin(theta)drd(phi)d(theta)...
The question I am dealing with has to do with the volume contained by two intersecting shapes I have created this integral and can't find a reasonable way of solving it. What is the best approach to solve this:
\int_{r=0}^{\pi/2}\int_{u=0}^{2\cos\theta} \sqrt{9-r^2} * r drdu=
I have this vector function:
\vec V = xe^{-r}\hat i
I have to obtain the volume integral:
I = \int(\vec \nabla \cdot \vec V)d^3x
What is that 'r' and how do I compute the volume integral?
"Find the volume of the region enclosed between the survaces z=x^2 + y^2 and z=2x"
I figured that the simplest way of doing this was to switch to a cylindrical co-ordinate system. Can someone check that the limits of integration are then
-\frac{\pi}{2}\leq \theta \leq\frac{\pi}{2}
0\leq\ r...
I'm have trouble trying to evaluate the volume integral (shown in question.gif).
I've attempted integrating it a few different ways, either achieveing an answer of 3 or 5.75, and I'm not sure where I'm going wrong. (Some of what I've done is in attempted_solution.gif)
Any comments...