Integrals Definition and 1000 Threads

What principles apply when making a substitution of variables in an improper integral. I gather that a substitution of variables can change an impoper integral to a proper integral. Can substitution also change a proper integral into an improper integral? Suppose I know that a pair of integrals...

Homework Statement Homework Equations The Attempt at a Solution Why I can't integrate\theta from 0 to 2\pi? Then integrate \varphi from 0 to \pi. It seems it can also generate a sphere.

Homework Statement I have a problem in part b Homework Equations The Attempt at a Solution How to integral the function?

I've been solving this exercise and I came to a point when one function can get two different integrals: Am I doing something wrong? Because both functions are the same, and the integrals (indefinite) are really different. This is a huge problem, because this is almost the final step of an...

Suppose I know the value of an integral: \int_0^T cos(\theta)dt = x Is there any way to evaluate the integral \int_0^T sin(\theta)dt solely from this information? EDIT: \theta=\theta(t), i.e. \theta is a function of t.

This is a continuum mechanics/fluid dynamics question concerning the time rate of change of a surface integral of a vector field, where the surface is flowing along in a velocity field (like in a fluid). (Gauss's law is for fixed surfaces.) This integral goes by various names in different...

Homework Statement \int (x+1)^2 dx Homework Equations The Attempt at a Solution I am just getting into this, and this is a simple problem, but my book and I took two separate routes. My question, essentially, is does any constant you get just "combine" with the "any constant" C...

Write a triple integral to represent the volume of the solid The wedge in the first octant and from the cylinder y^2 + z^2 <= 1 by the planes y=x, x=0, z=0 First.. i find the range for z..; 0 <=z<= sqrt(1- y^2) then... i find the range for y..; let z =0 0<=y<=1 next, if i...

What class will I start to study these?

Homework Statement Homework Equations The Attempt at a Solution I don't understand what exactly is going on here. They let u=(1+x^{2}), so that leaves them with this: \int \frac{x}{u^{2}}dx The derivative of (1+x^{2}) is simply 2x. And so: \frac{du}{dx} = 2x \rightarrow du =...

Hey there.. i try to solve the question below.. but.. i still didn't get the answer given by my lecturer.. the answer should be.. pi/4(e - 1) where did i do wrong? http://imageshack.us/photo/my-images/215/06072011697.jpg/ http://imageshack.us/photo/my-images/17/06072011699.jpg/...

hello,everyone i want to know how to solve this bessel function integrals: \int_{0}^{R} J_m-1(ax)*J_m+1 (ax)*x dx where J_m-1 and J_m+1 is the Bessel function of first kind, and a is a constant. thanks.

I am just starting to do double integrals and came acorss an issue. I remembered from single integrals when we integrate from limits for say -1 to 1, we can double it and change integration limits to 0 to 1. Now, when is this the case? Basically, when can we not do this?

I'm integrating 1/(z-1/2) over the closed disk w/ radius = 3 centered at 0. I've seen other problems where the final answer was i2pi times f(w) - here w =1/2. Since f(z) is equal to 1. Is the final answer just i2pi? Next up: I have the integral of dt/(2 + sint) the problem then tells me...

Homework Statement In an effort to keep me from spending all summer lying on the couch, I recently started reading Michael Spivak's Calculus on Manifolds; while working on problem 1-6 I got stuck on a technical detail and I was wondering if anyone could provide a little insight. Problem 1-6...

Homework Statement I've come across integrals of exponential and trig functions and I have no idea how to do them. Integration by parts doesn't really work because they just derive into either e or another trig function. One of them is \intsin(a)*sin(b - a)da Another is \inte(a)*sin(a)da...

Homework Statement \int_{C}(x+yz)dx + 2xdy + xyzdz C goes from (1,0,1) to (2,3,1) and (2,3,1) to (2,5,2) The Attempt at a Solution For C going from (1,0,1) to (2,3,1) x=1+t, y=3t, z=1; 0\leq t \leq 1 x'(t)=1, y'(t)=3, z'(t)=0 \int^{1}_{0}(1+t+3t)*1dt + 2(1+t)*3dt + 0...

Homework Statement I want to combine the 2 integrals: \int_{a}^{b}(x-3)f(x)dx+\int_{-b}^{-a}(x-3)f(x)dx Homework Equations given: f(x) is an even function The Attempt at a Solution swap the limits in the second integral: \int_{a}^{b}(x-3)f(x)dx-\int_{-a}^{-b}(x-3)f(x)dx...

Homework Statement int[sin(x)*cos(x)]dx Homework Equations U-substitution The Attempt at a Solution Alternative #1: u=sin(x) du=cos(x)dx intdu = u^2/2 + K = sin^2(x)/2+K Alternative #2 u=cos(x) du=-sin(x)dx -int[u]du = -u^2/2 + K = [U]-cos^2(x)/2+K From...

I'm on q4 of this paper: http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/2008/Paper63.pdf For the first bit i said the coordinates had ranges t \in ( - \infty , \infty) , \quad \chi \in [0, 2 \pi) Is that correct? Anyway for the next bit we can take the equation g_{\mu \nu} u^\mu u^\nu...

Homework Statement I have to calculate the following line integral \int_{\gamma}y^{2}cos(xy^{2})dx + 2xycos(xy^{2})dy where \gamma is the path defined by the equations x(t) = t^{4} and y(t)=sin^{3}(\frac{t\pi}{2}) t between 0 and 1Homework Equations Now I know that the formula for calculating...

If someone could link me to a tutorial on how to put in functions into a post, I would appreciate it, thanks. I am going to be putting in screen shots. Homework Statement http://img864.imageshack.us/img864/1517/scr1305133657.png" http://img864.imageshack.us/img864/1517/scr1305133657.png...

Homework Statement solve integral x^3/(e^x-1) with limits from 0 to infinity Homework Equations The Attempt at a Solution i tried using a rectangular contour,the boundaries of the contour pass through z=0 but the complex equivalent has pole at z=0. by Cauchy theorem the function...

Homework Statement Find the Volume of the given solid Bounded by the cylinders y^2+z^2=4 and x=2y, x=0,z=0 in the first octantHomework Equations double integral over a region D with f(x,y) dAThe Attempt at a Solution I graphed it in a xyz plane and got these intervals D = {(x,y)|...

Homework Statement Find surface integral of vector field F=<x,y,x+y> over the surface z=x^2+y^2 where x^2+y^2 less than 1. Use outward pointing normals Homework Equations The Attempt at a Solution So I did the whole thing and got a zero which doesn't look right to me. My algebra...

Should Say Intervals.. I'm tired... I am probably going wrong somewhere but I am running into problems with understanding this. My understanding of a 95% confidence interval is that in a sample of n the sample mean is 95% likely to be within 1.96 standard errors of the actual mean. I have a...

Homework Statement Given \mathbf{F} = \nabla f\; where \;f(x,y) = sin(x-2y) Find a curve C that is not closed and satisfy the equation \int_C \mathbf{F}\cdot dr = 0The Attempt at a Solution \nabla f = \;<cos(x - 2y),-2cos(x-2y)> So to satisfy the dot product being 0 (I am hoping I can do...

And why are the partitions not equal to one value? Why x1, x2, ... , xk, ... , xn-1, xn ? And why |the norm| -> 0 ? I was just curious if there is some specific logic behind it or if it is just there to discuss things in general. Thanks a lot. P.S.: Norm is the partition having the...

Homework Statement Prove the Mean Value Theorem for Integrals Proof Let f(x) be defined on [a,b] Let M be the max of f(x) and m be the min of f(x) Then m \leq f(x) \leq M \int_{a}^{b}m \;dx\leq \int_{a}^{b} f(x)\;dx \leq \int_{a}^{b} M\;dx m(b-a) \;dx\leq \int_{a}^{b} f(x)\;dx \leq...

Homework Statement Evaluate this integral using trigonometric substitution. \int_{0}^{2} \frac{x^3}{\sqrt{4-x^2}}dx Now I can do this the "textbook memorization" method like every calculus student does, but I want to go ahead an analyze this further. But I will show you the...

In various quantum chemistry books and course booklets I came across spin wave functions (usually referred to as alpha and beta for spin and up and down, respectively) that depend on a so called spin-variable. They are usually used to construct slater determinants. An example of this is Modern...

Homework Statement find volume of solid bounded by z=x, y=x, x+y=2 and z=0 The Attempt at a Solution first need to find domain. for x bounds, when y=0, x=0, when y = x, x+x=2 so x=1 therefore 0 < x < 1 for y bounds, x < y < 2-x now I am trying to work out what i integrate...

We are given a vector field: F=\frac{-y}{x^2+y^2} , \frac{x}{x^2+y^2} Then asked if F is conservative on R2 \ (0,0). I just solved the partial derivatives of each part of the vector field and they did indeed equal each other, but I don't under stand what the "\(0,0)" part means. We are then...

Hi i need to create a MATlab m file solving the following function for 0 to 90 degrees of \theta_0 and for any function F(\theta^'). [PLAIN]http://rogercortesi.com/eqn/tempimagedir/eqn9903.png [edit] dx should be d\theta^', sorry about that. I managed to do it in MATlab using symbolic maths...

I already posted about this but I redid the problem and got another answer: 1. Homework Statement The volume of an air mattress is 15 cubic feet. Air escapes at a rate of r(t)=0.25e^(−0.05t), where r is in cubic feet per second. Assuming the mattress is still completely inflated when the...

Homework Statement Evaluate the integral with respect to x from 0 to infinity when the integrand is x^2/(1+x^6), using complex integration techniques. Homework Equations The Attempt at a Solution I have no idea where to start. Please help!

Hello! I know that the theory of complex analysis is useful to compute integrals of real valued functions. I am a Physics student and I followed a Complex Analysis course but we did not have time to cover this up. I am looking for a textbook that takes a practical approach to this subject. I...

Hello dear colleagues! Yesterday i was trying to proof the surface area of a sphere formula, then i got some problems. I know that something is seriously wrong in this concept, but i can't tell what exactly is wrong. Could you guys help me please? I just thougt about a hollow sphere, then we...

Homework Statement It's a physics problem, where i have to evaluate the root-mean-square radius defined by the expression below. (First for a constant \rho, then for a "(r)dependent" \rho).Homework Equations (\int{_0}{^\infty} \rho r^4 dr / \int{_0}{^\infty} \rho r^2 dr) ^(1/2)The Attempt at a...

Homework Statement Use a triple integral to calculate the volume of the solid enclosed by the sphere x^2 + y^2 + z^2=4a^2 and the planes z=0 and z=a Homework Equations Transform to spherical coordinates (including the Jacobian) The Attempt at a Solution I'm stuck, as the radius...

Homework Statement \int\int\int^{}_{B} ye^(-xy) dV where B is the box determined by 0 \leq x \leq 4, 0 \leq y \leq 1, 0 \leq z \leq 5.Homework Equations The Attempt at a Solution \int^{4}_{0}\int^{1}_{0}\int^{5}_{0} ye^(-xy) dzdydx Integrating the first time I get zye-xy Plugging in 5 and 0 I...

Suppose that the bounded function f:[a,b]-->R has the property that for each rational number x in the interval [a,b], f(x)=o for all x in [a,b]. Prove that the lower integral of f from a to b is less than or equal to zero which is less than or equal to the upper integral of f from a to b...

I am trying to find the region between a surface z= x+4y and the region D in the x-y plane, where the region is the triangle with verticies (1,1) (2,3) (0,0).. However I am not sure how to come up with the double integral?

Homework Statement Find the centroid x,y,z of the region R cut out of the region 0<=z<=5sqrt(x2+y2) by the cylinder x2+y2=2x. Homework Equations x2+y2 = r2 x= rcosθ y= rsinθ The Attempt at a Solution Centroid x being Mx/m I'm guessing I've been working on this problem...

“Non-integrable” multiple integrals for Mathematica Dear all, I have been trying to crack one problem in Mathematica, but I keep getting a wrong answer probably because I have something either fundamentally wrong analytically or code wise. OK, here is the problem. Suppose we have to...

\int^{1}_{0}\int^{0}_{-x} \frac{ysin(pi*y^2)}{1+y} dydx Not exactly sue how to start this. I know that I need to integrate with respect to y first then use that solution and integrate again with respect to x however I do not believe integrating the initial problem is possible. Is there another...

#1 Homework Statement \int r^4 ( ln (r) ) drHomework Equations Infinity algebra and Calc related formulae..The Attempt at a Solution Not sure even where to start here.. I'm thinking a u-substitution, letting u = r^3 so that I can deal with the two left over r's, but I don't think that it would...

Homework Statement Find the volume of the cone bounded below by z=2root(x2+y2) and above by x2 + y2 + z2 = 1 Homework Equations The Attempt at a Solution Ok I have the solution, I just don't understand how to get it! So I know I have to change into spherical coordinates but...

Homework Statement Evaluate the double integral ∫∫D xy dA where D is the triangular region with vertices (0,0) (6,0) (0,1). Homework Equations The Attempt at a Solution 0 <= x <= -\frac{1}{6}x+1 0 <= x <= 6 the first integral would be the integral from 0 to -1/6x+1 of xy with...

find the integral int(1/z)dz along r for the curve: square with corners 1+i, -1+i, -1-i, 1-i traversed clockwise and anti-clockwise Homework Equations i know that clockwise will be the -(int) of the anticlockwise The Attempt at a Solution the first line = (1-2t)+i it's derivative...