Tagged Questions
2
votes
0answers
16 views
Given $\Sigma$ a surface parameterized by $\Phi : D \to \Sigma$, prove a certain formula for $area(\Sigma).$
Let $\Sigma$ be a surface parameterized by $\Phi : D \to \Sigma$, and let $$A=\Phi_u \cdot \Phi_u~,~B=\Phi_u \cdot \Phi_v,~ C=\Phi_v \cdot \Phi_v.$$ Prove $$area(\Sigma)=\int\int_D \sqrt{AC-B^2} ...
1
vote
1answer
64 views
A little help integrating this torus?
Let $\mathbf{F}\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by
$$\mathbf{F}(x,y,z)=(x,y,z).$$
Evaluate $$\iint\limits_S \mathbf{F}\cdot dS$$ where $S$ is the surface of the torus ...
2
votes
1answer
88 views
How to integrate $\cos\left(\sqrt{x^2 + y^2}\right)$
Could you help me solve this?
$$\iint_{M}\!\cos\left(\sqrt{x^2+y^2}\right)\,dxdy;$$
$M: \frac{\pi^2}{4}\leq x^2+y^2\leq 4\pi^2$
I know that the region would look like this and I need to solve it as ...
1
vote
1answer
27 views
How to determine a function of 2 variables from its derivative?
Please even the slightest advice would help!
If I have a function $V$ made of 2 variables $x_1$ and $x_2$,
and its derivative $$\frac{dV}{dt} = \frac{dV}{dx_1}\frac{dx_1}{dt} + ...
2
votes
1answer
68 views
Evalute this integral using Green's Thereom
Let C be the boundary of the half-annulus
$$1\leq(x^2+y^2)\leq4$$ where $$x\le0$$
in the xy plane, traversed in the positive direction.
Evaluate : $ \displaystyle \int_{c}(7\cosh^3(7x)-2y^3) ...
2
votes
3answers
134 views
How to solve this integral for a hyperbolic bowl?
$$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
0
votes
0answers
42 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( ...
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 ...
0
votes
1answer
81 views
How to calculate this integral?
Define
$$F=(x^2+y-4,3xy,2xz+z^2)$$
Compute the integral of Curl F over the surface $x^2+y^2+z^2=16, z\geq 0$
0
votes
1answer
78 views
Multivariable integral
What is the result of the following integral?
$$ 2 \cdot \int_0^{\infty} \frac{1}{\sqrt{2\pi s}}e^{-\frac{b^2}{2s}}
\int_{1-s}^{\infty}\frac{b}{\sqrt{2 \pi u^3}}e^{-\frac{b^2}{2u}} du db$$
where $0 ...
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 ...
1
vote
0answers
29 views
The tightest bound on an integral
Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following ...
0
votes
1answer
27 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?
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 ...
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
2answers
86 views
Find integral of a polar function $h(r,\theta)$ over a circle
I am studying for my math final and our prof gave us a review but without any solutions or hints. I don't really understand this problem so if anyone could help me out here I would appreciate it.
...
1
vote
0answers
74 views
Multivariable weird function
I have to prove two statements, but this function is so weird( and hard to work with )...I just can't figure out how to solve this.
The given function is $\varphi:\mathbb{R}^N\rightarrow\mathbb{R} $ ...
-1
votes
1answer
59 views
Moment of inertia of a circle
A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$
Thank you
2
votes
1answer
59 views
Work and Line Integral
A two-dimensional force field is given by the equation $$f(x,y)=cxy\textbf{i}+x^6y^2\textbf{j}$$, where $c$ is a positive constant. This force acts on a particle which must move from $(0,0)$ to the ...
0
votes
1answer
37 views
Triple Integral Spherical Coordinates
So I have to compute the triple integral of this:
$\int\int\int \frac{1}{1+x^2+y^2+z^2}$ and it says the equation of the sphere is $ x^2 + y^2 + z^2 = z$ which is just an elongated sphere running ...
2
votes
1answer
69 views
tricky surface integral
I am studying for my final and my prof gave us review questions but with no answers so I am lost with this question. If anyone can help I would really appreciate it.
Question: Find the area of the ...
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
75 views
Volume integral over a bounded region
Class is over now and I am studying for my final and I have a problem with this question on our review sheet. If anyone can help I would appreciate it.
Question: Find the volume of the region in ...
1
vote
1answer
32 views
Line integral of $F = r \times k$ on hemisphere
Exam revision -
Verify Stokes theorem directly by explicit calculation of the surface and line integrals for the hemisphere $r=c$, with $z \geq 0$, where $F = r \times k$ and $k$ is the unit vector ...
2
votes
2answers
38 views
Double Integral of piece wise function?
Let $I=[0,1]\times[0,1]$ and let $$f(x)= \begin{cases}
0, & \text{if (x,y)=(0,0)}\\
\frac{x^2-y^2}{(x^2+y^2)^2}, & \text { if (x,y)$\not=$(0,0)}\\
\end{cases}
$$
Need to show that ...
1
vote
1answer
62 views
Polar Coordinates: Dividing by the variable “r.”
Evaluate the iterated integral by converting to polar coordinates:
$\large \int^2_0 \int^{\sqrt{2x-x^2}}_0 xy~dy~dx$
I successfully completed most of the problem; however, I had difficulty ...
2
votes
1answer
27 views
volume evaluated by triple integral
Let $\Omega:=\{(x,y,z)|x^2+y^2=1, 0\leq z \leq 2\}$, fix an $\alpha \in (-\frac{\pi}{2},\frac{\pi}{2})$ and given the transoformation $T(x,y,z):=(x,y+z\tan \alpha,z)$, find the volume of $T(\Omega)$. ...
1
vote
1answer
49 views
How to integrate a vector function in spherical coordinates?
How to integrate a vector function in spherical coordinates?
In my specific case, it's an electric field on the axis of charged ring (see image below), the integral is pretty easy, but I don't ...
0
votes
2answers
43 views
Determining the Moment of inertia
Let $a,b,c$ be positive real numbers such that $c<a$. Suppose given is a thin plate $R$ in the plane bounded by $$\frac{x}{a}+\frac{y}{b}=1, \frac{x}{c}+\frac{y}{b}=1, y=0$$ and such that the ...
1
vote
2answers
42 views
Double integral of polar coordinates?
Compute $\int_C (8-\sqrt{x^2 +y^2}) ~ds$ where $C$ is the circle $x^2 + y^2 =4$.
Answer: $24\pi$
How is the answer $24\pi$? I converted the integral into a double integral of polar coordinates ...
2
votes
2answers
68 views
Integral from $0$ to Infinity of $e^{-3x^2}$? [duplicate]
How do you calculate the integral from $0$ to Infinity of $e^{-3x^2}$? I am supposed to use a double integral. Can someone please explain? Thanks in advance.
5
votes
0answers
97 views
Algorithm to calculate multiple integral.
One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
2
votes
3answers
69 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 $z=1$
...
3
votes
1answer
56 views
Sketch of the ordinate set of $f$
Let $f$ be defined on $[0,1] \times [0,1]$ as follows:
$f(x,y)= \begin{cases} x+y \mbox{ if } x^2 \leq y \leq 2x^2 \\ 0 \mbox{ otherwise} \end{cases}$
I want to make a sketch of the ordinate set of ...
4
votes
2answers
149 views
Double integral application
I need to determine $$\int_{0}^{1} \int_{-\sqrt{x}}^{\sqrt{x}}\frac{1}{1-y}dydx$$
I integrate in terms of the y component and obtained: $$\int_{0}^{1}\ln(\frac{1+\sqrt{x}}{1-\sqrt{x}})dx$$
Can ...
-3
votes
1answer
58 views
Property of double integrals
Let $f,g : A \rightarrow \mathbb{R}$ be integrable functions on a closed rectangle $A \subset \mathbb{R}^n$.
Show that $f+g$ is integrable and $\int_{A}f+g= \int_A f+ \int_A g$
Thank you
3
votes
2answers
114 views
Finding surface area of a cone
I will describe the problem then show what I tried to solve it.
I need to find the area of the cone defined as follows:
$$z^2=a^2(x^2+y^2)$$
$$0\leq z\leq bx+c$$
where $a,b,c>0$ and $b<a$.
...
1
vote
1answer
40 views
(Calculus 4) Compute the line integral with respect to s along the curve C.
I'm having a lot of trouble with this problem, and I suspect my mistake is somewhere in the setup. Here is the problem:
$$\int_C \frac{1}{1+x} ds$$
$$C: r(t) = ti + \frac{2}{3}t^{3/2}j, 0 \le t ...
3
votes
1answer
84 views
Show $g(\mathbf{x}) \leq h(\mathbf{x})$ implies $\int g(\mathbf{x})\mathrm{d}\mathbf{x} \leq \int h(\mathbf{x})\mathrm{d}\mathbf{x}$
Suppose I have $g$ and $h$ from $\mathbb{R}^p\to\mathbb{R}$ such that for all $\mathbf{x}$, $g(\mathbf{x}) \leq h(\mathbf{x})$. I want to prove that the integral over all $\mathbb{R}^p$ of $g$ is less ...
2
votes
3answers
57 views
Calculate volume in a 3D sort of space using cartesian coordinates
Find the volume bounded by the cylinder $x^2 + y^2 = 1$, the planes $x=0, z=0, z=y$ and lies in the first octant. (where x, y, and z are all positive)
1
vote
3answers
74 views
Change of variables in two dimensions
This is from Munkres' Analysis on Manifolds, Section 17, Question 4.
(a) Show that
$$ \int_\Bbb {R^2} e^{-(x^2+y^2)} = \left[ \int_\Bbb R e^{-x^2}\right]^2,$$
provided the first of these ...
0
votes
0answers
20 views
Riemann integrable then J-integrable
Let $E\subset\mathbb{R}^n$ be a closed Jordan domain and $f:E\rightarrow\mathbb{R}$ a bounded function. We adopt the convention that $f$ is extended to $\mathbb{R}^n\setminus E$ by $0$.
Let $\jmath$ ...
2
votes
0answers
35 views
Riemann integral is zero for certain sets
The question is:
Let $\pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}$. Prove that if $E\subset\pi$ is a closed Jordan domain, and $f:E\rightarrow\mathbb{R}$ is Riemann integrable, ...
0
votes
1answer
27 views
help taking line integral over a vector field
I have a problem in which I'm given a force field $\vec{F}(x,y,z)=x\hat{i}+y\hat{j}+ 3\hat{k}$ and a path $\vec{r}(t)=4cos(t)\hat{i}+4sin(t)\hat{j}+3t\hat{k}$ over the interval $0\le t\le 2\pi$. I ...
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 ...
0
votes
2answers
39 views
Surface Integral Q
I've been revising this area and I've completely forgotten what I'm doing and my notes are sketchy.
Evaluate $\int r \cdot dS$ over the surface of the sphere, radius a, centred at the origin.
...
