# Tagged Questions

0answers
31 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 ...
1answer
79 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 ...
1answer
25 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} + ...
1answer
60 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) ...
3answers
104 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)
0answers
36 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} ...
2answers
46 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( ...
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 ...
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 $x^2+y^2+z^2=16, z\geq 0$
1answer
77 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 ...
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 ...
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 ...
1answer
26 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 ...
1answer
37 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?
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 ...
2answers
53 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 ...
1answer
43 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 ...
2answers
82 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. ...
0answers
72 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 $\varphi:\mathbb{R}^N\rightarrow\mathbb{R}$ ...
1answer
56 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
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 ...
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 ...
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 ...
0answers
36 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 ...
2answers
74 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 ...
1answer
31 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 ...
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 ...
1answer
60 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 ...
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 $T(\Omega)$. ...
1answer
48 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 ...
2answers
42 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 ...
2answers
38 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 ...
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.
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 ...
3answers
68 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 ...
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 $z=1$ ...
1answer
55 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 ...
2answers
148 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 ...
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
2answers
113 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 $b<a$. ...
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 ...
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 ...
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)
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 ...
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 $\jmath$ ...
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, ...
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 ...
0answers
39 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 ...
2answers
36 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. ...
0answers
87 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 ...
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