Bounded Definition and 537 Threads

Homework Statement Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis: x+3=4y-y^2 x=0 About x-axis Homework Equations The Attempt at a Solution I tried solving x+3=4y-y^2 for y, but I can't get it even though it seems simple...

Hi, I would like to know if the function f(x) = (x+2)^-1 is bounded on the open interval (-2,2)? The interval doesn't include the point x = -2 but I'm not sure if I can say that there is a K>=0 such that |f(x)| < K for all x in (-2,2). The function is defined everywhere in that interval but...

Consider the region bounded by y = x^2, y = 5x+6, and the negative x-axis Compute the area of this region. Im somewhat confused by what they mean by the negative x-axis? The points of intersection between the two functions are [-1,1] and [6,36] A = Integral -1 to 6 (5x-6-x^2)? I'd...

Lets say we have a solution u, to the cauchy problem of the heat PDE: u_t-laplacian(u) = 0 u(x, 0) = f(x) u is a bounded solution, meaning: u<=C*e^(a*|x|^2) Where C and a are constant. Then, does u is necesseraly the following solution: u = integral of (K(x, y, t)*f(y)) Where K...

Equation of a hypocycloid is: x^{3/2} + y^{3/2} = a^{3/2}. Find the area of the figure bounded by this hypocycloid. My work: I can use the plane polar coordinates here taking x = a\cos t & y = a\sin t with t = [0, 2\pi]. But I don't know how to obtain the surface integral for evaluating the...

Find the area of a figure bounded by the equilateral hyperbola xy = a^2, the x-axis, and the lines x = a, b = 2a. My work: The equations of the lines and curves involved here are: xy = a^2 y = 0 x = a I don't know how b=2a is treated as an equation of a line here & hence I am puzzled as how to...

Let f be a real uniformly continuous function on the bounded set A in \mathbb{R}^1. Prove that f is bounded on A. Since f is uniformly continuous, take \epsilon = m, \exists \delta > 0 such that |f(x)-f(p)| < \epsilon whenever |x-p|<\delta and x,p \in A Now we have |f(x)| < m +...

I'm having trouble with this for some reason. If A:\mathcal{H}\to \mathcal{H} is a bounded operator between Hilbert spaces, the norm of A is ||A|| = \inf\limits_{\psi \neq 0} \frac{||A\psi||}{||\psi||}. My trouble is in verifying that ||A|| is in fact a bound for A in the sense that...

Could someone explain me how these three concepts hang together? (When is a set bounded but not closed, closed but not bounded, closed but not compact and so one?)

I have this problem bothering me. I am asked to find the volume bounded by two curves, when they were rotated about the y axis. I did it as usual. Functions are y1 = cosx +1 [tex] y_2 = 2(\frac{x - \pi}{\pi}) ^2[/itex] The way I did the problem is to find the volume of revolution of...

Suppose the universe is bounded and not infinite, would it require more dimensions for the universe to curve back upon itself? Like most, I find it impossible to picture 3 dimensional space having a boundary.

Hi. I'm new here. :) I was wondering if anyone could help me out with this problem... i'm supposed to find the region bounded by: y=x+1 y=e^-x x=1 i think i should find the other point of intersection but i forgot to do that (i haven't taken a math course for about 4 years). please help!

Some rule says that not all bounded sequence must be convergent sequence , one example is the sequence with general bound: Xn=(-1)^n could anyone help?! thanks in advance!

Prove that if f is uniformly continuous on a bounded set S then f is bounded on S. Our book says uniform continuity on an interval implies regular continuity on the interval, and in the previous chapter we proved that if a function is continuous on some closed interval then it is bounded...

hey Can someone please give me an example of an essentially bounded function?? I'm a bit lost.

If f is defined on [a,b] and for every x in [a,b] there is a d_x such that if is bounded on [x-d_x, x+d_x]. Prove that f is bounded on [a,b]. This question seems very odd. If every point, and indeed the neighbourhood of every point is bounded, then of course the function itself must be bounded...

Q: Show that every bounded set in R has a least upper bound. Using either "Every monotonic and bounded sequence is convergent" or "Every bounded sequence has an accumulation point" or "Every bounded sequence has a convergent subsequence" I'm not really sure how to start this out, but would...

I have absolutely no clue how to start here. Let F be the set of expressions of the form a = sum from i in Z of a-sub-i*x*i, where each a-sub-i is an element of R and {i < 0 : a-sub-i does not equal 0) is finite. (X is a formal symbol, not a number). An element a belonging to F is positive if...

"If I and J are bounded, then I\capJ is also bounded." Now, I was able to do this using the definition of suprema and infima and so fourth, but it is one godawful mess. I could sumbit it as is, but I was wondering if there's an easier way.

Find the volume of the solid region R bounded above by the paraboloid z=1-x^2-y^2 and below by the plane z=1-y The solution to this problem is: V=\int_{0}^{1} \int_{-\sqrt{y-y^2}}^{\sqrt{y-y^2}} (1-x^2-y^2)dxdy -\int_{0}^{1} \int_{-\sqrt{y-y^2}}^{\sqrt{y-y^2}}(1-y)dxdy I thought that...

I am having trouble finding the area between 2 polar curves... I have the procedure down, but the bounds are throwing me off. Any help with understanding how to bound would be great appreciated! I have attatched one problem that I am having hard time with and the work I have done. I know...

Q. Find the volume of the solid which is bounded by the cylinders x^2 + y^2 = r^2 and y^2 + z^2 = r^2. To me they don't really look like equations of cylinders, more like circles. Would the term "r" be constant in this case? Or would it be a variable? Even if r is a variable, I don't understand...

Can anybody help me find the volume y=sinx, bounded by the y axis, and the line y=1. The axis of revolution is the line y=1. I tried so many times and I can't find the correct answer, in the sheet it says (pie^2)/2 - 2pie

There are two problems I got stuck... 1. Is the area in the first quadrant bounded between the x-axis and the curve \ y= \frac{x} {2*(x^2+2)^{7/8}} finite? This one, I used the Area formula... but then I cannot integrate it... and then how to determine if it's finite or not? 2...

The homework question is this: Prove If D is a bounded subset of R then D bar = D U D’ is also bounded where D’ is the set of accumulation points of D. What is a general outline of a proof?

Here is the problem: First Part (already done): Find the volume of the solid that is bounded above by the cylinder z = 4 - x^2, on the sides by the cylinder x^2 + y^2 = 4, and below by the xy-plane. Answer: \int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 - x^2}}\int_{0}^{4 -...

Here is the problem: Find the volume of the solid that is bounded above by the cylinder z = 4 - x^2, on the sides by the cylinder x^2 + y^2 = 4, and below by the xy-plane. Here is what I have: \int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 - x^2}}\int_{0}^{4 - x^2}\;dz\;dy\;dx\;=\;12\pi...

Evaluate the integral over the bounded closed region B \int \int_{B} x^2 y^3 dx dy B is the region bounded by y=x^2 and y =x certainly the x=0 to x=1 will be the limits for the x part but what about the y does it go from root y to y? please help! i neeed to understand how to setup...

Find the area of the region bounded by: r= 6-2sin(\theta) here's what i did: 6-2sin(\theta) = 0 sin(\theta) = 1/3 so the bounds are from arcsin(-1/3) to arcsin(1/3) right? my integral: \int_{-.339}^{.339} 1/2*(6-2sin(\theta))^2 i get a answer of 0.6851040673*10^11, and it's...

How do I do this problem below? Plz guide me step by step or a least show me the correct bounds so I can learn...thanks ***Find the area bounded by y^2=x and y=x=2 Thanks so much

problem: find volume bordered by cylinder x^2 + y^2 = 4 and y+z=4 and z=4. the answer is said to be 16p. but I couldn't find it. I found it in double integral part.so it must be solved with double integral. I tried with Jacobian tranformation. nut still couldn't solve it. I was confused...

Find the area bounded by the two curves: x=100000(5*sqrt(y)-1) x=100000(\frac{(5*sqrt(y)-1)}{(4*sqrt(y))}) i'm having a lot of trouble trying to find the lower and upper limit of the two functions. I tried setting the two functions together and solving for y, but i get 0. then trying to...

could someone give me an example of a function that is bounded but is nonintegrable? i need to know what a nonintegrable function bounded on [a,b] is as said in my preperation file for a test? urgent help needed

If A is a bounded subset of the reals, show that the points infA, supA belong to the closure A*. At first the answer seems obvious to me since A* contains its limit points. I'm just having trouble putting it into words, any suggestions would be great, thanks.

This is the problem: find the area of the region bounded by the curves f(x) = x^2 + 2 and g(x) = 4 - x^2 on the interval [-2,2] I did the whole integral from -2 to 2 with (4-x^2) - (x^2 + 2) dx because the graph of g(x) is on top between the region bounded. But from my drawing, the points...

Can someone please help me w/ these problems below: Find the volume generated by revolving the area abounded by x=y^2 and x=4 about a)the line y=2 b)the line x= -1 ***I tried to write out the integral not sure if it's correct: a) V=pi* int. of (sqrt(x)^2-(2-sqrt(x))^2) dx **integral form...

I don't know if anyone will be able to help me, I am really stuck on this question! "Show that the volume of the region bounded by the cone z=sqrt((x*x)+(y*y)) and the parabloid z=(x*x)+(y*y) is PI/6" The bits in the brackets (ie x*x and y*y) are x squared and y squared respectively and...