Tagged Questions
1
vote
1answer
42 views
Integral sign with circle (AND arrow on the circle) through it
I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path.
So when I encountered in complex analysis the above integral sign but with ...
2
votes
3answers
93 views
Vector calculus for ellipse in polar coordinates
I'm having trouble with this question, can somebody please help me with it! I'll thanks/like your comment if help me =)
I know that for a ellipse the parametric is $x=a\sin t$ , $b= b \cos t$, ...
1
vote
1answer
66 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
1
vote
2answers
76 views
Find the area of $A = \{ \langle x,y\rangle \in \mathbb{R}^2 \mathrel| (x+y)^4<a x^2 y,\ x>0 \}$?
I can't really think of how to set the limits
2
votes
0answers
33 views
Closed curves question
Can you give me some help on the following problem?
Given two closed curves $\alpha, \beta : \mapsto \mathbb{R}^3$ we define $\phi_{\alpha \beta}: I^2 \mapsto \mathbb{R}^3$ as $\phi_{\alpha \beta} ...
0
votes
0answers
44 views
separating a variable from integral
In the following integral, I would like to separate $\alpha$ from rest of the equation. Can we solve the following equation for $\alpha$?
$$\large{\int_{0}^{a} \int_{0}^{2\pi} ...
1
vote
2answers
48 views
Computing $\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$ using substitution
Consider this integral:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$$
How would you compute it?
I already solved this problem this way:
$$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz = \left( ...
1
vote
3answers
55 views
Using Spherical coordinates find the volume:
Inside the surfaces $z=x^2+y^2$ and $z=\sqrt{2-x^2-y^2}$
I integrated over the ranges:
$0 \leq \theta \leq 2\pi$
$ 0 \leq \phi \leq \frac{\pi}{2}$
$0 \leq r \leq \sqrt{2}$
I get ...
0
votes
1answer
54 views
Theorem or just a change of varibles?
I have a formula in my text:
$$\int \int_{S} F \cdot n dA= \int \int_{w} F(G(u,v)) \cdot (dG_{u}\times dG_{v}) du dv$$
I am really lazy and hate remembering formulas to me this looks like a ...
2
votes
1answer
46 views
When is it valid for me to just integrate a trig function?
I am having a problem identifying when I need to use some kind of integration technique or am I just over complicating things. Could someone please explain to me when I need to or not?
Normally, I ...
-2
votes
1answer
45 views
Integrate the function $f(x, y, z)=\sqrt{3x^2 + 3y^2 + z + 1}$ over the surface given by…
Integrate the function $f(x,y,z) = \sqrt{3x^2 + 3y^2 + z + 1}$ over the surface given by the graph of $z = g(x,y) = x^2 + y^2$ over the region $1 \leq x^2 + y^2 \leq 4$
2
votes
2answers
37 views
Is there a need for another integration technique?
I'm being asked to calculate
$$I\triangleq\int_0^1\int_{e^{\large x}}^e{xe^y\over(\ln y)^2}\,dy\,dx\quad.$$
I got stuck on the indefinite inner one,
$$J\triangleq\int{e^ydy\over(\ln y)^2}\quad.$$
At ...
0
votes
1answer
28 views
Finding volume under surface and above a region
I'm asked to find $\underset{U}{\int}(x+y)^2\, dA$ where U is a region bounded by the lines
x = -1, x = 1, y = -1
... and by the curves
x=$y^2$ , y=1+$x^2$
Plot: http://d.pr/WYSg
I started out by ...
1
vote
1answer
38 views
Finding the centre of mass? What axis does the centre of mass lie on?
Let the mass density $\mu$ be given by
$$\mu(x,y,z)=
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq1$$
what axis would the centre of mass lie on?
2
votes
1answer
46 views
A parameterized elliptical integral (Legendre Elliptical Integral)
$$
K(a,\theta)=\int_{0}^{\infty}\frac{t^{-a}}{1+2t\cos(\theta)+t^{2}}dt
$$
For $$ -1<a<1;$$ $$-\pi<\theta<\pi$$
I know this integral to be a known tabulated Legendre elliptic integral, ...
0
votes
4answers
51 views
Contour Integral help with residue theorem
$$
K = \int_{0}^{\infty}\frac{1}{x^{4}+x^{2}+1}dx
$$
I am supposed to use contour integration to solve this, but I can't even determine the singularities. The denominator doesn't have any that I can ...
0
votes
3answers
35 views
Confusing Triple Integral
i'm having trouble with this integral
the integral is $\int_0^9\int_{\sqrt z}^3\int_0^y z\cos(y^6)\,dx\,dy\,dz$.
We aren't given any more information and i'm a bit stuck as to where to start. I don't ...
1
vote
2answers
54 views
Triple integral problem involving a sphere
Let $R = \{(x,y,z)\in \textbf{R}^3 :x^2+y^2+z^2\le\pi^2\}$
How do I integrate this triple integral
$$\int\int\int_R \cos x\, dxdydz,$$ where $R$ is a sphere of radius $\pi$?
I have trouble ...
1
vote
3answers
75 views
surface area of a sphere above a cylinder
I need to find the surface area of the sphere $x^2+y^2+z^2=4$ above the cone $z = \sqrt{x^2+y^2}$, but I'm not sure how. I know that the surface area of a surface can be calculated with the equation ...
0
votes
1answer
44 views
Integration in $\mathbb{R}^n$ region
If its all parameterized usually I can solve it, but I have a problem with integration in vagues regions, usually I dont know the right procedure to solve them.
The problem I need to solve is: given ...
0
votes
1answer
42 views
Directional derivative and Laplace operator
Let $R$ be a region in the plane satisfying Green's theorem, with boundary $C$. Let $S$ be an open set containing $R$, and let $\phi$ be a scalar field $\phi: S \rightarrow \mathbb{R}$ with continuous ...
2
votes
0answers
37 views
Intuitive understanding of integral of vector valued functions
Today in class we were introducing complex line integrals. And that got me thinking, I don't know of a good interpretation for integrals of functions from $\mathbb{R}$ to $\mathbb{R}^2$ or ...
1
vote
2answers
85 views
Cylindrical coordinates where $z$ axis is not axis of symmetry.
I'm a little bit uncertain of how to set up the limits of integration when the axis of symmetry of the region is not centered at $z$ (this is for cylindrical coordinates). The region is bounded by ...
1
vote
0answers
44 views
Finding the average value of a function over region in $\mathbb{R}^3$.
I want to know if I set this multiple integral up correctly (I always mess them up!). I want to find the average value of $z$ over the region (call it $M$) bounded by $x^2+y^2+z^2=16$ and ...
0
votes
2answers
68 views
What is wrong with this approach?
I am given a problem to find the surface area of the cylinder $z^2+y^2=9$ above the rectangle defined by the points $(0,0),(4,0),(0,2),(4,2)$.
Instead of trying to integrate on with any respect to ...
0
votes
1answer
42 views
Change of Limits When Changing Order of Integration
When you change your order of integration, you have to change the limits. The way I've been taught is to draw a graph in order to find the new limits. Is there a formula to find the new limits of ...
1
vote
1answer
71 views
Stokes theorem problem to find alpha and beta so that I is independent of the choice of S
I have a question that I got half through but can't finish it. If anyone could help I would appreciate it.
Question: let C1 be the straight line from (-1,0,0) to (1,0,0) and C2 the semi circle ...
1
vote
1answer
34 views
Solving differential equation for x
I have a field $\phi(x,t)=\sin(t+|x|)(\frac{x}{|x|})$ where x is a point vector and t is the current time.
If this field describes the acceleration of a particle at a point in space and time:
...
2
votes
1answer
29 views
parametric integration
Is there a mistake in the bottom of page 5 of this document? INTEGRATION: THE FEYNMAN WAY
$$\frac{\partial}{\partial b} e^{be^{ix}}= e^{ix}e^{be^{ix}}$$ instead of $ib e^{be^{ix}}e^{ix}$? thank you ...
0
votes
0answers
53 views
Find area of a curvilinear triangle that includes hyperbolic functions
We were given this question in class and I tried to compute it and it looks to e pretty crazy. Can anyone take a look and let me know if I did it correctly... I would really appreciate it.
...
1
vote
2answers
92 views
Find area of a simple, smooth, closed curve lying in a plane
I was given this question in class and I assume it is a spin off of Green's theorem for finding the area of a closed curve $\lambda$ in 2D but expanded to 3D I believe. Anyways I am pretty confused ...
2
votes
3answers
70 views
Really Confused on a surface area integral can't seem to finish the integral off.
Basically the question asks to compute $\int \int_{S} ( x^{2}+y^{2}) dA$ where S is the portion of the sphere $x^{2} + y^{2}+ z^{2}= 4$ and $z \in [1,2]$ we start with a chnage of variables
$x=x ...
2
votes
3answers
52 views
Find the volume of the region contained above $z=1$ and below $x^{2}+y^{2}+z^{2}=4$
Why doesn't this work?
Find the volume of the region contained above $z=1$ and below $x^{2}+y^{2}+z^{2}=4$
going to cylindrical this should be easy. $z=(4-r^{2})^{\frac {1}{2}}$ and $z=1$
...
2
votes
3answers
76 views
Algebraic way to change limits of integration of a double integral
I know how to graphically change the limits of integration of a double integral. That is, by graphing the region and eyeballing (a.ka.a "looking at") it to determine the new limits. But an answer to a ...
1
vote
1answer
46 views
Find the center mass of right circular conic shell of base radius $a$ and height $h$ and constant density
We were assigned the question for homework:
Find the center of mass of a right circular conic shell, radius $a$, height $h$ and constant density $\rho$.
This is a multi-variable calculus class and we ...
0
votes
0answers
47 views
Finding marginal distribution from joint distribution when domain of one variable is given in terms of the other variable
Given distributions $f_X(x)$ and $f_{Y \mid X}(y \! \mid \! x)$:
My aim is to find $f_Y(y)$.
I find the joint density as
$$ f_{XY}(x,y) = f_{Y \mid X}(y \! \mid \! x) f_X(x) = 0.5(y-x) \quad ...
2
votes
3answers
128 views
Integration with Spherical Coordinates
Use spherical coordinates to find the volume of the solid inside both $x^2+y^2+z^2=16$ and $z=(x^2+y^2)^{1/2}$.
2
votes
0answers
36 views
Finding a surface area over an inconvenient region
So I was digging in my old notes and found this teaser from a few months ago that neither the class nor the teacher could solve:
Find the surface area of the part of $z=4-x^2-y^2$ that lies above the ...
1
vote
0answers
40 views
Line and surface integrals $R^{3} $
So i actually missed the class where this material was covered so plaese bear with me if my understanding is not so good. one of the problems in my textbook is as follow's.
Prove the following ...
1
vote
0answers
88 views
Tough integration with change of variables and switch to polar coordinates
I was given this question in class and I was just wondering if I am on the right track…
Evaluate:
$$I=\iint\left(1-\frac{x^2}{a^2} -\frac{y^2}{b^2} \right)^{3/2} dxdy $$ over the region enclosed by ...
0
votes
2answers
215 views
Multiple integration to find the moment of inertia of a cylinder.
I was assigned a question asking: Find the moment of inertia of a circular cylinder with base $a$ and height $h$ about the diameter of the base. We are working through applications of multiple ...
3
votes
1answer
102 views
$\iiint e^{-x^2-2y^2-3z^2}dV$
I was given this question in class but I don't understand how to do it.
Evaluate the triple integral in $\mathbb{R}^3$ of $\iiint e^{-x^2-2y^2-3z^2}dV$.
The hint was to use the idea that $\int ...
3
votes
2answers
221 views
Find volume inside the cone $ z= 2a-\sqrt{x^2+y^2} $ and inside the cylinder $x^2+y^2=2ay$
I have this question and I have seen here that there are similar questions but I have been trying to figure it out for a while with no luck.
I have to find volume inside the cone $z= ...
1
vote
1answer
77 views
How to find limits of integration on a convolution of CRVs
In finding the convolution of two independent and continuous random variables, I am struggling with limits of integration. I cannot seem to figure out over what intervals the probability density ...
1
vote
1answer
56 views
$e^{F(x,y)}$ Type Multi-variable Exponential Integrals
I am sure all you integration buffs can do this faster than I can type it. Your help with a quick explanation and solution is appreciated.
$$F _{XY} = \int_0^\infty\int_0^\infty xye^{-\frac{x^2 + ...
1
vote
0answers
39 views
Change of dependent variables
I am hoping to evaluate the integral
$$ \int_0 ^R u(r,t)r^2\,dr$$
However, I do not have an expression for $u$, but rather $\tau(r,t)$, where the two are related by
$$\frac{\partial\tau}{\partial ...
5
votes
3answers
59 views
Triple Integral and symmetry
The problem is as follows: Compute the intergal
$$I=\iiint_B \frac{x^4+2y^4}{x^4+4y^4+z^4} \:dx\:dy\:dz,$$ where $B$ is the unit ball defined by $B=\{(x,y,z) \mid x^2+y^2+z^2 \leq 1\}$.
The official ...
0
votes
1answer
71 views
Finding the winding number of a curve
Let $\gamma(t)=(r \cos t,r \sin t)$, for some $r>0$, and let $\Gamma$ be a $C^2$-curve in $\mathbb R^2-\{\bf 0\} $, with parameter interval $[0,2 \pi]$, with $\Gamma(0)=\Gamma(2 \pi)$, such that ...
4
votes
0answers
64 views
Stokes' Theorem Example
I am reading Wade's Introduction to Analysis. One of the exercises is to show that
$$
\int_{\partial M}\sum_{k=1}^n \, dx_1dx_2\cdots \hat{dx_i}\cdots dx_n
$$
is equal to the volume of $M$ if $n$ is ...
3
votes
1answer
345 views
Baby Rudin, chapter 10, problem 1 - independence of order in multiple integrals.
In problem 1, Rudin asks for a generalization of example 10.4, which states that if $f$ is continuous in the standard k-simplex $Q^k$, then the integral $\int_{Q^k} f$ exists, and that the order of ...
