Bounded Definition and 514 Threads

Homework Statement Using a suitable Jacobian, evaluate the volume bounded by the surface ##z = 2 +x^2##, the cylinder ##x^2 + y^2 = a^2## (where ##a## is a constant), and the ##x-y## plane. Homework Equations ##x = r cos{\theta} ## ##y = r sin{\theta} ## The Attempt at a Solution I...

Checking my steps and answer. Thanks in advance! Compute \int_0^3 \int_0^2 \int_1^3 xyz\ dz\ dy\ dx. \int_0^3 \int_0^2 \frac{xyz^2}{2} \Big|_1^3 = \frac{9xy}{2}-\frac{xy}{2} = \frac{8xy}{2} = 4xy \int_0^3 2xy^2 \Big|_0^2 \int_0^3 8x\ 4x^2 \Big|_0^3 = 36

Would someone like to have a conversation with me about the bounds of the universe and energy? I have a few ideas rambling around.. When I say bounds I mean the expanding bubble. But I would like to discuss infinite matter and infinite energy big crunch possibility. I'm a beginner with BIG thoughts.

Homework Statement The area bounded by the loop of the curve ## 4y^2 = x^2(4-x^2) ## is in sq. units 7/3 8/3 11/3 16/3 Homework Equations NA The Attempt at a Solution By putting x = 0 and x = 2 I am getting y = 0. Getting complex y values after x exceeds 2. I am not getting where the loop...

Hi All, Question: Consider the tetrahedron, T, bounded by planes x=2, y=0, z=0 and 3x-6y-2z=0. Determine the integral \iiintyDV which is the y coordinate of the centre of mass. I am getting a negative area which leads me to believe I'm doing something wrong. Working is attached. Help would...

i m thinking of this... area bounded by x=0, y=0, y=x^2, y=4-x^2 and x=2 why the region bounded by the below three cases are the same 1. x=0, y=0, y=x^2, y=4-x^2 2. y=0, y=x^2, y=4-x^2 3. and x=0, y=x^2, y=4-x^2 but after i add x=2 and compute the bounded region, it's different? i am just...

Homework Statement Let C be the composition operator on the Hilbert space L_{2}(\mathbb{R}) with the usual inner product. Let f\in L_{2}(\mathbb{R}), then C is defined by (Cf)(x) = f(2x-1), \hspace{9pt}x\in\mathbb{R} give a demonstration, which shows that C does not have any eigenvalues...

Homework Statement Integrating ##^{\frac{\pi}{2}}\int_{\frac{-\pi}{2}}(1-u^{2})^\frac{1}{2}u_{x}dx##, and using the result : ##\int(1-u^{2})^{\frac{1}{2}}=\frac{1}{2}u(1-u^{2})^{\frac{1}{2}}+\frac{1}{2}arcsin(u)## Homework Equations I'm pretty sure it is just the integral itself were I am...

Homework Statement a) sketch the region in the first octant bounded by the elliptic cylinder 2x^2+y^2=1 and the plane y+z=1. b) find the volume of this solid by triple integration. Homework EquationsThe Attempt at a Solution I have already sketched the elliptic cylinder and the plane. my...

Hi guys I am very new here this is my second post. (sorry in advance i don't know how to use the functions of the site fully yet) i think this is the correct method to follow, some feedback or hints would be great thanks in advance! 1. Homework Statement Find the area bounded by where...

Homework Statement find the volume using spherical coordinates of the region bounded above by z=9 and below by z=sqrt(x^2+y^2) in the first octant. Homework EquationsThe Attempt at a Solution I found this volume using cartesian and cylindrical coordinates, so I know the answer I am looking...

I've been very confused with this proof, because if a sequence { 1, 1, 1, 1, ...} is convergent and bounded by 1, would this be considered to be a Cauchy sequence? I'm wondering if this has an accumulation point as well, by using the Bolzanno-Weirstrauss theorem. I really appreciate the help...

Find the centroid of the region bounded by y = 1.5x2 − 14x + 23.5, and x − y = 8. You should enter the coordinates of your answer either as decimals or fractions. Your answer must be accurate to within 0.001. I got a wrong answer of (7/3, 59/52), I'm having problems plugging in the numbers. Any...

Hey! :o I am looking at the proof of the theorem that for any rectangle the outer measure is equal to the volume. At the beginning of the proof there is the following sentence: It is enough to look at the case where the rectangle R is closed and bounded. Why does it stand?? (Wondering) Is...

Homework Statement Courant states that a convergent sequence is necessarily bounded; that is, for all n, the absolute value of term an is less than or equal to some number M. My question is does this apply to the sequence given by an = 1/(n-1)? Homework Equations As n approaches infinity, an...

Hello, Let's suppose we are given a function f:\mathbb{R}\rightarrow \mathbb{R}, and we assume its Fourier transform F=\mathcal{F}(f) exists and has compact support. What sufficient condition could we impose on f, in order to be sure that F is also bounded?

Homework Statement Let R be the region in the first quadrant bounded by all three of the curves x = 2, y = 1, and y = (x−4)^2. Compute the volumes V1, V2, and V3 of the solids of revolution obtained by revolving R about the x-axis, the y-axis, and the x = 5 line, respectively. FIRST, I...

Homework Statement Find the optimal trajectory x*(t) that minimizes: J = \int_{0}^{1} \left( \frac{\dot{x}(t)^2}{2} + 3x(t) \dot{x}(t) + 2x^2(t) + 4x(t) \right) dt with x(0) = 1 and x(1) = 4 Homework Equations Euler's equation: \frac{\partial g}{\partial x} -...

Homework Statement Hi I require to compute the volume of a ellipsoid that is bounded by two planes. The first horizontal (xy) plane is cutting directly along the mid-section of the ellipsoid. The second horizontal plane is at a z = h below the first horizontal plane. The volume of the...

Must double integrate using type I or type II planar region D to find volume bounded by Cylinder y^2+z^2=4 And Planes X=2y X=0 Z=0

For $a>0$, let $V$ be the volume created by revolving the region bounded by $y=b\sqrt{x}$ and $x=a$ around the axis $x=a$. The units of $x$, $y$, and $a$ are $[m]$. The units of $b$ are $[m^{1/2}]$. The value of $b$ remains fixed. For fun and practice, I did this question both with cylindrical...

Homework Statement The problem and solution are attached as TheProblemAndSolution.jpg. Homework Equations Definition of the limit of a sequence. The Attempt at a Solution I understand how P = ϵ + |A| can be seen as an upper bound that proves that the sequence is bounded, but for the last bit...

Let $f:[0,\infty)\to\mathbb R$ be a differentiable function such that for all $a>0$ exists a constant $M_a$ such that $|f'(t)|\le M_a$ for all $t\in[0,a]$ and $f(t)\xrightarrow[n\to\infty]{}0.$ Show that $f$ is uniformly continuous. Basically, I need to prove that $f$ is uniformly continuous...

Hello again I have another question regarding absolute min-max over a region. This is a weird one. My function is: \[f(x,y)=x^{2}+y^{2}-xy\] and the region is: \[\left | x \right |+\left | y \right |\leq 1\] Now, I have plotted the region using Maple: The answer in the book where it...

If ##\left\{ a_{n} \right\}## is monotone increasing and there exists ##M \in \Re## such that for every ##n \in N## ##a_{n} ≤ M## prove that ##\left\{ a_{n} \right\}## converges. (Hint: Use the Cauchy sequence property. Recall: 1) ##\left\{ a_{n} \right\}## is Cauchy if and only if...

Hi, I just need these solutions checked. Thank you in advance! Consider the region bounded by the following curves ##y=x-3, y=5-x, \text{and}\ y=3##: 1.) set up an integral expression that would give the area of the region of y as a function of x: ##y = x-3 = 5-x## ##x + x - 3 -...

Homework Statement Find the area of the region in the first quadrant, which is bounded by the x-axis, the line x = 2 and the circular arc x^2 + y^2 = 8Homework Equations The Attempt at a Solution I didn't use the hint given in the question but does my answer still makes sense. Did I set up the...

is it possible to work backwards from a spectrum to which operator?

Can some show me how we show a LTI system is BIBO? I read the definition but it didn't help. For example, how would we show if \[ H(s) = \frac{s - 2}{(s + 2)(s + 1)(s - 1)} \] is BIBO stable or unstable?

It seems strange, but would a metric space consisting of two points, X={a,∞} be totally bounded, but not bounded? because d(a,∞)=∞. But for all ε>0, X=B(ε,a)UB(ε,∞). It's been proven that totally bounded→bounded, so this is wrong. Why?

Hey again! :) Let the sequence $(a_{n})$ with $a_{1}>0$ and $a_{n+1}=1+\frac{2}{1+a_{n}}$.Show that the subsequences $a_{2k}$ and $a_{2k-1}$ are monotonic and bounded.Find the limit $\lim_{n \to \infty} a_{n}$,if it exists. Do I have to show separately that the two subsequences are monotonic and...

Hello, quick question really. Homework Statement Find the area bound by the x axis, x = 1, x = 4 and y = 2/x Homework Equations The Attempt at a Solution Representing this graphically, the question is equivalent to performing the definite integral of y = 2/x from 1 to 4. Right? Which...

Homework Statement The figure shows a semicircle with radius 1, horizontal diameter , and tangent lines at and . At what height above the diameter should the horizontal line be placed so as to minimize the shaded area http://imgur.com/grrCqWF Homework Equations The equation of a...

So it's been a while since I've done one of these problems. Need to make sure I am using the right procedures to solve it. Q)Find the area bounded by the curve $y = \frac{1}{2}x^2$ and $x^2 + y^2 = 8$ So first thing I did was plug in numbers to get the two graphs. It looks like they intersect...

This is a concise question, so the title pretty much says it all. Also, this is not a HW question, but the idea has subtly popped up in two homework problems that I have done in the past. I cannot justify why the entire set would be bounded, because we know nothing of the nature of the...

Suppose I have B: X\to Y and A: Y\to Z, where X,Y,Z are Banach spaces and B\in \mathcal L(X,Y) and A\in \mathcal L(Y,Z); that is, both of these operators are bounded. Does it follow that AB \in \mathcal L(X,Z) and \| AB \|_{\mathcal L(X,Z)} \leq \|A\|_{\mathcal L(Y,Z)} \|B\|_{\mathcal L(X,Y)}...

Homework Statement It asks to find the volume of the solid given these planes: z = x y = x x + y = 2 z = 0 It also asks to find the volume using 2 iterated integrals with different orders of x and y integration. Homework Equations The Attempt at a Solution I found...

Hello. Please look over my answers! Homework Statement a) Prove that this set is not convex: x = [1, 2] U [3, 4] c R b) Prove the intersection of two bounded sets is bounded Homework Equations for a) x = [1, 2] U [3, 4] c R The Attempt at a Solution a) A convex set is where...

How can it be shown that $$16x\cos(8x)+4x\sin(8x)-2\sin(8x)<|17x|?$$ This problem arises from work with damped motion in spring-mass systems in Differential Equations. I have gotten to this inequality after some algebraic manipulation, but am completely stuck here. Here is the illustrative...

Homework Statement Prove that a subset in R^n, where n is a finite number, is bounded if and only if it is totally bounded.Homework Equations If A is the subset, A is bounded if there is a point b in R^n such that d(x,b)<= K, for a every x in A. A is totally bounded if for every e> 0, there...

Homework Statement Let G be a finite complex matrix group: G \subset M_{n\times n}. Show that, for g \in G, |\text{tr}(g)| \le n and |\text{tr}(g)| = n only for g = e^{i\theta}I. 2. The attempt at a solution Since G is finite, then every element g \in G has a finite order: g^r = I for some...

Sorry if this question seems too trivial for this forum. A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds. Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc...

Here is the question: I have posted a link there to this thread so the OP can view my work.

Homework Statement . Let ##X## be a complete metric space and consider ##C(X)## the space of continuous functions from ##X## to ##\mathbb R## with the metric ##d_{\infty}##. Suppose that for every ##x \in X##, the set ##\{f(x): f \in C(X)\}## is bounded in ##\mathbb R##. Prove that there exist...

I am reading Apostol's section on Riemann-Stieltjes integral and I have doubts on one statement: Let ##α## be a function of bounded variation on ##[a,b]## and suppose ##f \in R(α)## on ##[a,b]##. We define ##F## as ##F(x)=\int_a^x f(x)dα## if ##x \in [a,b]##, then ##F## is a function of...

Homework Statement . Prove or disprove that the function ##f(x)= x^2sin^2(\dfrac{\pi}{x})## if ##0<x\leq 1## and ##f(x)=0## if ##x=0## is of bounded variation. The attempt at a solution. I've seen the graph of this function on wolfram and for me it's clearly not of bounded variation since it...

Find the area bounded by the curve $$x = 6x - x^2$$ and the y axis. So can I use the even function rule to get: $$2 \int^2_0 6x - x^2 dx$$ I just need someone to check my work. $$2 [ 3x^2 - \frac{1}{3}x^3 ] | 2, 0$$ $$ 2 [ 12 - \frac{8}{3}]$$ $$2 [ \frac{36}{3} - \frac{8}{3} ]$$ $$2 * 28/3$$...

Hi everyone, :) Here's a question with my answer, but I just want to confirm whether this is correct. The answer seems so obvious that I just thought that maybe this is not what the question asks for. Anyway, hope you can give some ideas on this one. Problem: Let \(X\) be a finite...

Find the area bounded by the curve $$x = 16 - y^4$$ and the y axis. I need someone to check my work. so I know this is a upside down parabola so I find the two x coordinates which are $$16 - y^4 = 0$$ $$y^4 = 16$$ $$y^2 = +- \sqrt{4}$$ $$y = +- 2$$ so I know $$\int^2_{-2} 16 - y^4 dy$$...

Here is the question: I have posted a link there to this thread so the OP can see my work.