Integration Definition and 1000 Threads

Recently I started seeing integral calculus and right now we are covering the topic of the antiderivative. At first sign it was not very difficult, until we started seeing integral variable substitution. The problem starts right here: Let's suppose that we have a function like this: \int...

Homework Statement The question involves finding the arc length of the parametric equation x = e^t + e^-t and y = 5 - 2t Homework Equations Arc length of a parametric equation ∫√(dy/dt)^2 + (dx/dt)^2 dt limits are from 0<t<3 The Attempt at a Solution Taking the derivative of both x and y...

Homework Statement Find the area enclosed by the curve x = t^2 -2t, y = t^0.5 and the y axis Homework Equations Area of a parametric curve = ∫g(t) f'(t) dt, where g(t) = y and f(t) = x The Attempt at a Solution I believe that the limits of integration by be found by setting x and y equal to...

I would really appreciate it if people could help with these integrals. We are supposed to be doing integrals with this table here: http://math.boisestate.edu/~wright/courses/m333/IntegralTablesStewart.pdf Here are the two integrals. Technically, I only need one of them completed...

Hi - I have just started 'Computational Physics' by Koonin & Meredith, - through distance learning. Exercise 1.3 needs a program to evaluate an integral - I'm finding myself kinda rusty on integrals. The hint says - split range of integration into parts, use different change of variable in each...

Homework Statement F[/B]ind the Antiderivative of: (x^3-1)/(x-1). All is known is the integration formulas (i.e. ∫sinx = -cosx+c) Homework Equations Integration Formulas the most complicated being ∫cscx dx= -ln(cscx+cotx)+c The Attempt at a Solution I tried doing (x^3/x-1) -(1/x-1), but now...

I don't understand the relationships between the integration limits of Maxwell Equations (specifically the ones in integral form in matter) Is this related to Stokes/Gauss' Theorems? or something else?

I have a question concerning how how we define the differentiation and integration operators. Firstly, I know that functions are typically defined as an ordered triple triple ##(X, Y, f)## such that ##f⊆X×Y##, where ##x \in X## and ##f(x) \in Y##. This all seems nice and fine, but we also define...

Greetings :) Well I wanted to seek help if my solution is on the right path, given as follows: 1) \int cos ^2x dx So my solution follows like this: u = cos^2x du = 1/2 (1+cos(2x)) v = x dv = dx but I've stuck when its in the u.v - \int v.du cos^2 (x) - \int...

Hello. Let's imagine that we have a parametric function f1(x(t),y(t),z(t)) and an analytic one f2(x,y,z) and we have to integrate their product over some volume dx dy dz. So what are analytical tools for it? Thanks!

Homework Statement Find the residue of: $$f(z) = \frac{(\psi(-z) + \gamma)}{(z+1)(z+2)^3} \space \text{at} \space z=n$$ Where $n$ is every positive integer because those $n$ are the poles of $f(z)$Homework EquationsThe Attempt at a Solution This is a simple pole, however: $$\lim_{z \to n}...

There is a nice equation made by Nobuo Yamamoto which describes the curve of an egg and it is: (x^2 + y^2)^2 = ax^3 + (3/10)xy^2, where a is the length of the major axis of the egg. Solve this equation for y, we get: y=+/- sqrt((3/20)ax - x^2 + xsqrt((7/10)ax + (9/400)a^2)) When I rotate the...

Hi Attached is an extract of a paper by Lord Rayleigh on pressure generated during collapse of a bubble in a liquid. Will someone please explain how the RHS of equation (2) in the attachment is obtained ? TIA

I have asked the same question on math stackexchange under the moniker "anonymous," since I do not wish to be known there. I will try my luck here.$$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ I don't understand, we say: $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Then we say: $$I =...

Homework Statement $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Homework Equations Below The Attempt at a Solution $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ I don't understand, we say: $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Then we say: $$I = \int_{-\infty}^{\infty} e^{-t^2} dt$$...

Hello, I passed by this integration and couldn't understand the moving from the left hand to the right hand side. $$ \int_{0}^{1/n}f(t)dt=\frac{1}{n}f(0) $$ could you please tell me why this is??

Homework Statement $$ \int x^{3}cos(x^{2})dx$$ The attempt at a solution OK, so I am aware that there is a way in which to do this problem where you do a substitution (let $$u=x^{2}$$ to do a substitution before you integrate by parts), and I was able to get the answer right using this method...

I am reading Spivak's Differential Geometry Vol. 1. I am stuck for some days in chapter 8 about integrating forms on manifolds. Maybe someone can clear my doubt. First, I will 'type' what the corollary says: My doubt is regarding this affirmation: The book it says is easy to see. Well...

http://en.wikipedia.org/wiki/Jefimenko's_equations What is the integral in these equations called? how do you integrate over (d^3)r'?

Homework Statement This is from Apostol's Calculus Vol. 1. Exercise 1.15, problem 6.(c) Find all x>0 for which the integral of [t]2 dt from 0 to x = 2(x-1) Homework Equations [t] represents the greatest integer function of t. The Attempt at a Solution [/B] Integral of [t]2 dt from 0 to x...

Homework Statement I have been trying to evaluate an integral that has come up in the process of me solving a different problem, but am completely stuck. As I have confirmed with Wolfram Alpha that the integral once solved yields the correct solution to my problem. However, I am trying to...

I posted the same question on Math Stackexchange: http://math.stackexchange.com/questions/1084724/calculating-harmonic-sums-with-residues/1085248#1085248 The answer there using complex analysis is great. I had questions, which Id like to get advice on here. (1) How did he get the laurent...

Hello, I am looking to evaluate: $$I = \int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2} dx$$ I will use a rectangular contour. The image looked weird here so the upload of the image is here: http://i.stack.imgur.com/W4BfA.jpg $R$ is more like the radius of the small semi circle, we have to let $R \to...

Several authors state the formula for finding the arc length of a curve defined by ##y = f(x)## from ##x=a## to ##x=b## as: $$\int ds = \int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx$$ Isn't this notation technically wrong, since the RHS is a definite integral, and the LHS is an indefinite integral...

Homework Statement The question is to get Fourier sine series of e^-x =f(x) on 0<x<1 Homework Equations Bn = 2/L ∫ (e^-x) * sin(nπx/L) over the limits 1 to 0, where L = 1 f(x) = summation of Bn*sin(nπx/L) The Attempt at a Solution So I integrated ∫ by part integration so I took u =...

The integral given below is to be computed as a function of real variables x and s. Even a partial answer only for s>0 is very useful. Here is the integral: $$\int_{0}^{\infty}{dk \frac{k^2 e^{-k^2 x^2}}{(k^2 + s)^{3/2}}}$$ Thank you for your help.

This is an interesting complex analysis problem; **The figure on the bottom left is what is being referred to,Fig7-10.** **Firstly: (1)** How is the branch point $z=0$ at $z=0$?? We have $f(0) = 0$ that is not a discontinuity is it? **Secondly:(2)** It says that: $AB$ and $GH$ are coincident...

In my computational physics textbook, three different velocity estimators are derived for a problem with equation of motion: \ddot x = F(x) where the positions are found by using the Verlet algorithm: x(t+h) = 2 x(t) - x(t-h) + h^2 F[x(t)] The three velocity estimators are: v(t) = \frac{x(t+h)...

Hello, Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ We know that because $f(x)$ is even:$$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = \frac{1}{2} \cdot \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Consider a complex function, with $z = x + iy$ $$f(z) =...

Hi, does anyone know how to integrate e^-x (sin(nπx))? I have tried part integration but that goes on until infinity... and I am not sure how to use the substitution method...Please help! I have tried taking e^-x as U but then I end up getting the entire canceled off then...

Homework Statement Which one is correct? ##\int (3x+2) (2x+1)^{\frac{1}{2}} dx = \frac{1}{3} (3x+2)(2x+1)^{\frac{3}{2}} - \frac{1}{15} (2x+1)^{\frac{5}{2}} + C## or ##\int (3x+2) (2x+1)^{\frac{1}{2}} dx = \frac{1}{3} (3x+2)(2x+1)^{\frac{3}{2}} - \frac{1}{5} (2x+1)^{\frac{5}{2}} + C## ...

Homework Statement The problem is f(x) = sin2πx - (1/πsquare)*sinπx and its given Bn sin (nπx) = f(x) Question is find Bn. Homework Equations Bn = 2/L ∫ (sin2πx - (1/πsquare)*sinπx) * sin(nπx/L) where L is 1 The Attempt at a Solution I did [/B] ∫ sin2πx * sin (nπx) - (1/πsquare)*sin...

Homework Statement So it says solve this wave equation : [y][/tt] - 4 [y][/xx] = 0 on the domain -infinity<x<infinity with initial conditions y(x,0) = e^(-x^2), yt(x,0) = x*(e^(-x^2)) Homework Equations I used the D Alembert's solution which is 1/2(f(x+ct)+f(x-ct)) + 1/2c ∫ g(z) dz The...

I'll like to know the probability density function for one of the x or y axis, given that there is an exponential decay of a material in two-dimensional space. So, that means I have to marginalize, say y and keep x, but I couldn't solve the integration. I even tried with Mathematica and Matlab...

Homework Statement So I'm calculating the gravitational potential of a sphere at at point P. R = radius of sphere, r = distance from center of sphere to point P. I'm looking at two scenarios; r > R (1) and r < R (2). So I have the following integral: \begin{equation} V(r) = \int...

I'm trying to practice for my final. The sample problem is: "Find the volume of the solid generated when the region bounded by y = x4and y = x1/3, 0<=x<=1, is revolved about the x-axis." To start, I set the two y equations equal to each to find the points of intersection. x4 = x1/3, : raise...

Homework Statement I'm checking to see if the momentum operator is Hermitian. Griffiths has the solution worked out, I'm just not following the integration by parts. Homework Equations int(u dv) = uv - int(v du) The Attempt at a Solution I've attached an image of my work. It seems there...

Homework Statement A program finds the area under the Gaussian Distribution Curve between ±σ using Simpsons Rule. Modify the program to investigate the effect of the number of strips. Do this by using a DO loop in the main program for the following sequence of number of strips (n); n-2, n-4...

Homework Statement Solve: ##x \frac{dy}{dx} - 4y = x^6 e^x.## Solution: Dividing by ##x##, we get the standard form ##\frac{dy}{dx} - \frac{4}{x}y = x^5 e^x.## Hence the integrating factor is ##e^{\int -\frac{4}{x}dx}= e^{-4 \int \frac{1}{x}dx} = e^{-4 \ln x} = e^{\ln x^{-4}} = x^{-4}##...

I was wondering if there is a convenient way of checking if a substitution is correct or not. For example, I tried solving for ∫(1/(a^2-x^2)dx using two different substitutions, x=acosu and x=asinu giving different solutions. I got the integral as arcsin(x/a) using x=asinu and -arccos(x/a) using...

Homework Statement Consider a simple model of a free-standing dam, depicted in the diagram. Water of density ρ fills a reservoir behind the dam to a height h. Assume the width of the dam (the dimension pointing into the page) is w. (a) Determine an equation for the pressure of the water as...

Homework Statement Using \int_{-\infty}^{\infty}e^{-x^2/2} dx = \sqrt{2\pi}, Integrate x^(5/2) e^(-x) dx from 0 to infinty 2. The attempt at a solution I tried substituting x = u^2/2 but i could not simplify further. Please help me with the problem. Thank you in advance.

Homework Statement A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes x = d and x = -d. The y- and z- dimensions of the slab are very large compared to d and may be treated as essentially infinite. Homework...

Evaluate ∫[sin2x/(1+(cos)^2 x) dx]Differentiate f(x) = (sin)^2 (e^((sin^2) x)) Hello, I'm just really stumped with these review questions and i have a test coming up. For the first, I'm not too sure what to do since there is a sin2x in general and for the second i don't know how to deal the...

I have a question why everyone says ∫uv' dx=uv-∫u'v dx why don't they replace v' with v and v with ∫vdx and say ∫uv dx=u∫vdx-∫(u'∫vdx) dx i think this form is a lot simpler because you can just plug in and calculate, the other form forces you to think backwards and is unnecessarily complicated.

Homework Statement sketch the solid region contained within the sphere, x^2+y^2+z^2=16, and outside the cone, z=4-(x^2+y^2)^0.5. b) clearly identifying the limits of integration, (using spherical coordinates) set up the iterated triple integral which would give the volume bounded by the...

Homework Statement ##(e^y + 1)^2 e^{-y} dx + (e^x + 1)^3 e^{-x} dy = 0## Homework EquationsThe Attempt at a Solution ##(e^y + 1)^2 e^{-y} dx + (e^x + 1)^3 e^{-x} dy = 0## ##(e^{2y} + 2 e^y + 1) e^{-y} dx + (e^{3x} + 3e^{2x} + 3e^x + 1) e^{-x} dy = 0## ##(e^{2y - y} + 2 e^{y - y} + e^{-y}) dx...

I have not taken maths so you may find my question silly. in physics i have to deal with integration.so can you please tell me where we write integration constant and where we don't?

My wave function: ##\psi_2=N_2 (4y^2-1) e^{-y^2/2}.## Definition of some parts in the wavefunction ##y=x/a##, ##a= \left( \frac{\hbar}{mk} \right)##, ##N_2 = \sqrt{\frac{1}{8a\sqrt{\pi}}}## and x has an arrange from ##\pm 20\cdot 10^{-12}##. Here is my integral: ##<x^2> =...

Homework Statement -h^2/2m (sqrt(2b/pi)) e^(-bx^2) d^2/dx^2 (e^(-bx^2)) dx from - to + infinity Homework Equations I tried differentiating e^(-bx^2) twice and it came up weird , I positioned the values and finally cam up with (-2b sqrt(pi/2b)...is there any other way to do it ? The...