Integration Definition and 1000 Threads

This is going to sound like a silly question, but here we go anyway! I've always thought about a definite integral being used for modelling a change in some quantity whilst an indefinite integral is employed to find the defining function of that quantity. For example, consider the...

I remember being given a ghastly book of integrals to learn when I was about 16. I went to sleep. Apparently the first book of integrals was published by Meier Hirsch in 1810. There have been many more since then. Surely with the invention of the internet there is something better? Symbolab has...

I have seen several functions be integrated by multiplying by a form of one or by adding a form of zero. When is it advantageous do do one of these things? Are there any example problems (Calc I or II) in which I can try these techniques?

In case of overhanging beam with point load at the end. For example: (here RA-reaction is negative) The equation will be as follows (by double integration method): , as we can see the equation will not have Point load (10kN) term in it. 1) How the influence of the point load is accounted in...

I have a gaussian charge distribution, in gaussian units $$ \rho(\mathbf r) = q\frac{\alpha^3}{\pi^{3/2}}\exp( -\alpha^2 r^2 ) $$ and I want to solve Poisson's equation to find the electrostatic potential $$ \nabla^2 \psi(\mathbf r) = -4\pi\rho(\mathbf r). $$ Since the charge distribution has...

There's one odd way to think about integration when it comes to interpreting it as a sum. suppose for a second that ##x## is in meters, you could think of distance as an infinite number ##n## of "points" in space, ##n→∞##, then in this case ##f(x)Δx## would mean that you now have ##nf(x)##...

Hi, This is my first question here, so please apologise me if something is amiss. I have two curves such that Wa = f(k,Ea,dxa) and Wb = f(k,Eb,dxb). I need to minimize the area between these two curves in terms of Eb in the bounded limit of k=0 and k=pi/dx. So to say, all the variables can...

Attempt at solution: Writing the chain rule for ## E(V,T) ##: ## dE = \frac{\partial E}{\partial T}dT + \frac{\partial E}{\partial V}dV ## Then, integrating the differential: ## \int{ dE } = \int{ \frac{\partial E}{\partial T}dT } + \int{ \frac{\partial E}{\partial V}dV } ## If I put the...

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So I can push this integral all the way to the end and see I get a negative volume. I solve for the intercepts of the cone and sphere at r^2 = 1/2. Seeing this cone is inside the sphere and the sphere is around it, I figure I should integrate from sqrt(1/2) to 1 since we're dealing with a unit...

Here, width of first bar, y=x^2=a^2 y=x^2=(a+Δx)^2 height of nth bar=y=(a+(N-1)Δx)^2 Total area,I={a^2+(a+Δx)^2+(a+2Δx^2)+...+[a+(N-1)Δx]^2}Δx I={Na^2 + 2aΔx +...} I can't seem to get forward to get the required result which is 1/3(b^3-a^3)

I'm calculating key rate (R^Rate-wise) by integrating R(eta) over all possible eta from 0 to 1, with a probability distribution (PDTC) which is a log-normal distribution. The equation of log-normal distribution: The equation of R(eta): Therefore, R^Rate-wise =...

Hi all, I have nuclear magnetic resonance spectrum. The vertical axis is intensity, and the horizontal axis is index. I need to find integral under the peak. But I am not sure, what region should I choose for integration - region 1 or region 2? Please find attached the spectrum.

(i) Dividing the rod into thicknesses of dx we get discs of area A with lengths=dx so using (****) we have the resistance of a typical disc (between point x' and x'+dx) as: (1) ##R(x'dx)=\frac{dx}{g(x)A}## (ii) Using (1) and (*) and the integrating from a to b of the entire rod we get...

##\int_{-\infty}^{z}sin(2kz)\,dz=\dfrac{-1}{2k} \Big [cos(2kz) \Big ]_{-\infty}^z=-\dfrac{cos(2kz)}{2k}+\dfrac{cos(-\infty)}{2k}##. I ended up here, and I don't know how to proceed. One recommended me to use contour integration, but I have no idea about it.

Let $S^{n-1} = \left\{ x \in R^2 : \left| x \right| = 1 \right\}$ and for any Borel set $E \in S^{n-1}$ set $E* = \left\{ r \theta : 0 < r < 1, \theta \in E \right\}$. Define the measure $\sigma$ on $S^{n-1}$ by $\sigma(E) = n \left| E* \right|$. With this definition the surface area...

This is my working out, and I also included the correct answer in the last line. The answer used a different method, however, what did I do wrong with my method? Thanks for the help!

Double integration maybe?? I calculated potential due to shell on plate's center but not on other points on it's surface.

Hello everybody, I am currently working on an experiment investigating the formation of planets. I have a vacuum chamber in which dust particles form bigger agglomerates through accretion (sticking together). From the imagery I can see those agglomerates which are build up by smaller...

This is the integral I try to take. ##\int\sqrt{1+9y^2}## and ##9y^2=tan^2\theta## so the integral becomes ##\int\sqrt{1+tan^2\theta}=\sqrt {sec^2\theta}##. Now I willl calculate dy. ## tan\theta=3y ## and ##y=\frac {tan\theta}3## and ##dy=\frac{1+tan^2\theta}3## Here is where I can only...

I want to compute: $$\oint_{c} F \cdot dr$$ I have done the following: $$\iint_{R} (\nabla \times v) \cdot n \frac{dxdy}{|n \cdot k|} = \iint (9z-1) dxdy$$ I don't know what limits the surface integral will have. Actually, I am not sure what's the surface. May you shed some light...

Hello, I'm just starting Zee's QFT in a Nutshell, I'm a bit confused about what he means by "integate by parts under the d4x". Can someone explain what he means by this? I understand how to obtain the Klein-Gordon equation from the free particle Lagrangian density, but not sure why he invokes...

<Moderator's note: Moved from a technical forum and thus no template.> > The tank (hemisphere) is full of water. Using the fact that the weight of water is 62.4 lb/ft3, find the work required to pump the water out of the outlet. The radius of the hemisphere is 10. ##V =\pi x^2 h## using the...

Homework Statement I need calculate the points (##x_i##) and weights (##w_i##) with Gauss Lobatto seven points on the interval [a,b]. With the points and the weights I am going to approximate any integral at this interval.Homework Equations I have found the relevant points and weights at the...

Finding $$\lim_{n\rightarrow \infty}\sqrt{n}\int^{\frac{\pi}{4}}_{0}\cos^{2n-2}(z)dz$$

Homework Statement 1) Calculate the density of states for a free particle in a three dimensional box of linear size L. 2) Show that ##\int f \nabla g \, d^3 x=-\int g \nabla f \, d^3 x## provided that ##lim_{r \rightarrow \inf} [f(x)g(x)]=0## 3) Calculate the integral ##\int...

##\mathbf{M'}## is a vector field in volume ##V'## and ##P## be any point on the surface of ##V'## with position vector ##\mathbf {r}## Now by Gauss divergence theorem: \begin{align} \iiint_{V'} \left[ \nabla' . \left( \dfrac{\mathbf{M'}}{\left| \mathbf{r}-\mathbf{r'} \right|}...

I'm trying to figure out this volume integral, a triple integral, of a 9-variable function. 3 Cartesian-dimension variables, and 6 primed and un-primed co-ordinates. After the volume integration, the un-primed co-ordinates will have been gotten rid of, leaving a field function in terms of...

Evaluation of $\displaystyle \int \sqrt{\frac{\sin^2 x-3\sin x+2}{\sin^2 x+3\sin x+2}}dx$

In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...

Homework Statement Integrate: $$\int \frac{dx}{x^2\sqrt{4-x^2}}dx$$ Homework EquationsThe Attempt at a Solution I got to the final solution ##\int \frac{dx}{x^2\sqrt{4-x^2}}dx=-\frac{1}{4}cot(arcsin(\frac{1}{2}x))##. But It's the method where you transform that to the solution...

So the classical law of force given by Newton is F= ma = dp/dt = qE. Thus if i integrate the last two equivalents I get: ∫(dp/dt)dt = q∫Edt p + C = q∫Edt correct? then what would the integral of...

Hi guys, i'm trying to find the velocity profile for a laminar flow in a round pipe. Starting from a force balance, we can obtain the first equation high in the left. I started with a procedure but i think I'm making mistakes. Can you suggest me the mathematical procedure?

Hello all. I'm using Griffiths' Introduction to Quantum Mechanics (3rd ed., 2018), and have come across what, on the face of it, seems a fairly straightforward principle, but which I cannot justify to myself. It is used, tacitly, in the first equation in the following worked example: The...

I tried learning calculus using the book by Spivak.In this text, while introducing integrals the author explained a lot about partitioning the area under the curve and defined the integral.The way I understood this is, as we increase the number of divisions in the partition the lower sum and the...

When estimating an integral using trapezoidal approximation, we can find the error or uncertainty in the estimation by: ##Error~in~T_n \leq \frac{M(b-a)^3}{12n^2}## where ##M## is the maximum value of the absolute value of f''(x) over [a,b], ##n## is the number of intervals, and ##T_n## is the...

When solving differential equations the following scripture can arise, for example: $$\int \frac{df}{\sqrt{\sin(\theta)^2-f^2}}$$ If the change of variable ##f=\sin(\theta)\sin(u)## Is performed, do the letters ##f,\theta## shall be considered independent or is...

This question is about the general 1 loop correction to the propagator in QFT (this is actually not important for this question). Let's say we have an integral over an integration variable x, and this x ranges from ##-\infty## to ##\infty##. If we look at this integration contour in the complex...

Homework Statement We solved the differential equation (2.29), , for the velocity of an object falling through air, by inspection---a most respectable way of solving differential equations. Nevertheless, one would sometimes like a more systematic method, and here is one. Rewrite the equation...

Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...

Let's say we have ##df=2xy^3dx + 3x^2y^2dy## - this is an exact differential. In integrating, to find f, can we write ## f = \int 2xy^3 \, dx + \int 3x^2y^2 \, dy = 2x^2y^3 + C ## Or am I getting it wrong?

I have a few of integration equations and need to convert it into Python. The problem is when I tried to plot a graph according to the equation, some of the plot is not same with the original one. The first equation is the error probability of authentication in normal operation: cond equation...

Homework Statement "Find ##\int_0^x \frac{6\ dt}{2t+1}##." Homework Equations ##\int \frac{\ dx}{x} = \ ln(x)## for ##x>0## The Attempt at a Solution Method 1: Let ##u=2t+1##, ##\frac{1}{2} du=\ dt##. ##\int_0^x \frac{6\ dt}{2t+1}=3\int_1^{2t+1} \frac{du}{u}=3\ ln(u)|_1^{2t+1}=3\ ln(2t+1)##...

I am modeling some dynamical system and I came across integral that I don't know how to solve. I need to integrate vector function f=-xj+yi (i and j are unit vectors of Cartesian coordinate system). I need to integrate this function over a part of spherical shell of radius R. This part is...

Homework Statement ##f(t) = -e^{-t}## ; ## t ≤ 0## and ##f(t) = 0## ;## t > 0 ## find Laplace transform this function. Homework Equations Laplace transform ##F(s) = \int_{[-∞<r<+∞]} f(t) e^{-st} dt## The Attempt at a Solution ##F(s) = \int_{[-∞<r<0]} -e^{-t} e^{-st} dt +\int_{[0<r<+∞]} (0)...

Homework Statement Please see attached image for the full scope of the problem, and to see the work drawn out by the text. My question lies with line 3 as it is clear that u-substitution was used on a definite integral, but the limits of integration were not changed. Homework EquationsThe...

Hello, I was struggling with solving a specific integral. I know that I can rewrite the exponential matrices and the range of the three Euler angles. However, I am not sure I should I write in terms those three Euler angles.

Well here is my (I hope successful this time ) attempt at 10. e) We consider the function ##f(x,y)=\begin{cases}\dfrac{e^{-x|y|}}{y}, y\neq 0 \\ 0, y=0\end{cases}##. For this function for ##y\neq 0## it is for any ##x##, ##\frac{\partial f}{\partial x}=-\frac{|y|}{y}e^{-x|y|}##. Also it...

Hi PF! The following function is long but only 3 command lines: first defines the function ff, second numerically integrates the function, and third plots the function. As you'll see the integral is zero yet clearly that is not the case (seen from the plot). Any idea what's happening? ff =...

Homework Statement acceleration of moving particle is described by a=-kv^1,5 where k is a constant. if the condition when t=0 is v=v0 and x=0 prove that xt = √(vv0).t Homework Equations dv/dt=a, dx/dt=v The Attempt at a Solution dv/dt=a dv/v^1,5=-k dt v^-1,5 dv = -k dt ← integrating both...