Integral Definition and 1000 Threads

I have this little doubt because in class, the professor said the work is equal to the integrate of vdP, but i don't know how to prove it, why it isn't PdV like those exercises in a cylinder-piston

Hello, I am studying arc lengths and areas for parametric curves from the Adams & Essex Calculus book and I am a bit baffled by example 2 in the image attached. I understand the solution in the book where they integrate from t=0 to t=pi/2 (first quadrant) and multiply by two to get the full...

c *************** BLOCK 2: SURFACE CARDS ************** 10 PZ 100 110 SO 110

Dear all, I have a rather basic question about an equation in David Tong's lecture notes on cosmology; see http://www.damtp.cam.ac.uk/user/tong/cosmo.html My question is about eqn. 1.19 (page 14), in which the cosmological redshift is derived. It's not about the physics, but about some basic...

Mentor note: Moved from a technical math section, so missing the template. TL;DR Summary: I'm trying to solve the differential equations (D^2 -4D + 3 )y = 2xexp(3x) + 3exp(x)Cos2x On this page , I've solve particular integral by 2 ways, 1st is above the line in which ( see 7th equality from...

I am looking at ## \int \tan^n dx ## where ##n## is a positive integer. The index ##n## has been reduced by writing ##\tan^n x ## as ##\tan ^{n-2} \tan^2 x## which is quite clear with me. We have, ## \int \tan^n xdx = \int \tan^{n-2} x⋅ \tan^2 x dx=\int \tan^{n-2}x ⋅(\sec^2 x -1) dx ##...

##\mathbb{Q}(\zeta_{p^\infty})##, also written as ##\mathbb{Q}(\mu_{p^\infty})## or ##\mathbb{Q}(p^\infty)##, denotes ##\mathbb{Q}## adjoined with the ##p^{n}##th roots of unity for all ##n##. It's the union of a cylotomic tower, and it's studied in subjects like Iwosawa theory and class field...

Let ##(E,\mathcal A)## be a measurable space equipped with a measure ##\mu##. If ##f:E\to\mathbb R## is integrable, then we have ##\left|\int f\,\mathrm{d}\mu\right|\leq\int |f|\,\mathrm{d}\mu##. If ##f:E\to\mathbb C## is integrable, Le Gall in his book Measure Theory, Probability and Stochastic...

Just looking at the summand, I can see that the function is ln(pi/4 + x^2) as the (i pi/2n) term is the 'x' term. How do I determine the limits of the integral, however? I was thinking about using the lower bound of the summation --> this given the (pi / 2n)^2 term, implying that nothing was...

I would like to evaluate expressions with Simpy, but unfortunately I am unable to get a simple answer, the one I would get by hand if I had the time to perform all the computations. As far as I understand, Mathematica does it and yields 4 times the Simpy result, which is a big worry since I wish...

I got the two relations for spherical and rectangular coordinates. In rectangular...

Hi friends, Can anyone offer some insight into this challenging integral? I can't seem to think my way through this. Thank you Stevesie $$ \int_{0}^{\infty}\frac{1}{x}\exp\left(-\frac{1}{2}\left( \frac{\log\left( x \right)-\mu}{\sigma}\right)^{2} \right)\exp\left(-\frac{1}{2}\left( \frac{ x...

I've got this integral I'm trying to find: $$ \int \frac{d \theta}{ \sqrt{1 - \cos \theta}} $$ To me it smells like trig sub, so I investigate the right triangle: Such that: $$ \cos u = \sqrt{1-cos \theta} $$ we also have from the same triangle: $$ \sin u = \sqrt{\cos \theta} $$ Square...

Hello frens, How should one approach this sort of integral? Any tips would be appreciated. Let's say we have $$ \int_{(1)}^{(2)}\exp\left[ a+b\exp\left[ f(x) \right] \right]dx$$ ...where the limits of integration are not important. Any tips? Thanks!

When I take ##x = 2\cos(t)## and ##y = 2\sin(t)##, the integral becomes ##\int_{t=\frac{\pi}{2}}^0 4(2\cos(t))^2 \cdot 2 dt = -8\pi##. The final answer is ##8\pi##. Why is my method wrong? I played around with desmos and the parameterisation seems correct...

I'm a little thrown off with material I'm going through right now. I already covered the whole "area under the curve" and using that to determine the volume of a given equation, but I'm confused now as to why calculating the surface area has a different method with ds? For example, say there...

Dear everyone, I have a question on how to show that an integral is divigent. Here is the setup: Suppose that we have the following function ##\sigma(x)=\frac{1}{x^{2-\varepsilon}}## for an arbitrary fixed ##\varepsilon>0.## \begin{equation}...

i started to think to maybe do an integral to find the minimum area, and then I thought that the area itself is not sufficient because there is more material depending on the slope. so I thought to do an integral depending on the length instead of x. ##dh^{2}=dx^{2}+dy^{2}## ##\int{}f(x)dh=...

I have ##1-x^2 = 1- \sin^2 θ = \cos^2 θ## and ## dx =cos θ dθ## ##\int_0^{0.5} (1-x^2)^{1.5} dx = \int_0^{\frac{π}{6}} [cos ^2θ]^\frac{3}{2} dθ = \int_0^{\frac{π}{6}} [cos ^4θ] dθ## Suggestions on next step.

First, we rewrite the term ##|\vec r-\vec r_q|## in the following way: $$|\vec r-\vec r_q|= \sqrt{(\vec r-\vec r_q)^2} = \sqrt{\vec r^2 + \vec r_q^2 -2\vec r\cdot\vec r_q} = \sqrt{r^2 + r_q^2 -2rr_q\cos\theta}$$ Due to rotational symmetry, we go to spherical coordinates: $$\phi_{e;\vec r_q} =...

Solving the integral is the easiest part. Using spherical coordinates: $$ \oint_{s} \frac{1}{|\vec{r}-\vec{r'}|}da' = \int_{0}^{\pi}\int_{0}^{2\pi} \frac{1}{|\vec{r}-\vec{r'}|}r_{0}^2 \hat r \sin{\theta}d\theta d\phi$$ then: $$I = \dfrac{1}{|\vec{r}-\vec{r'}|}r_{0}^2(1+1)(2\pi)\hat...

Howdy all, Let's say we have, in general an expression: $$ \int f(x) g(x) dx $$ But in through some machinations, we have, for parameter ##a##, $$ \int f(x) g(x) dx = \int f(x) g(a) dx $$ ...can we conclude that ## g(x) = g(a) ## ???? Thanks

Given that $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ and $$f(x)=e^x$$, is it true that $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$?

Given that $$\int_a^b f(x)g(x) \, dx = \int_a^b f(x)h(x) \, dx$$ and $$f(x)=e^x$$, is it true that $$\int_a^b g(x) \, dx = \int_a^b h(x) \, dx$$?

Hi If i calculate the definite integral between the limits of L and 0 of sin(nπx/L)sin(kπx/L) using the trig formula 2sinAsinB = cos (A-B) - cos (A+B) it is undefined when n=k because (n-k) appears in the denominator. If i calculate the same integral with n=k using the formula sin2(nπx/L) = (...

I'm trying to calculate the volume of a truncated hypersphere. As part of it I want this integral. Clearly when x=1 the integrand is zero. But plugging this into the series give me a number greater than one. It is true that the series is not defined for x=1, but subtracting some tiny sum...

Hello everyone, hope you are all well. I have the following problem: I have a temperatur-time graph. If you determine the integral of this graph, you get the unit [kelvin*second]. This unit is as far as I know meaningless. Is it possible to mathematically "transform" the area under the curve...

I am looking for a closed form solution to an integral of the form: $$ \int_0^\infty \frac{e^{-Du^2t}u \sin{ux}}{u^2+h^2} du $$ D, t, and h are positive and x is unrestricted. I have tried everything, integration by parts, substitution, even complex integration with residue analysis. I've...

I have come up with a solution, however, I'm not sure whether I'm correct. A fellow student of mine has a different result. I'm gonna show my solution, and hopefully one of you can confirm my result or tell me what I did wrong. $$ \begin{align} p_z &= \int d^3x z \rho(\vec{x}) \notag \\ &=...

We use the invariance of the measure under ##p\rightarrow -p## to get $$-\int d^3p\xi^{rT}\mathbf{p}\mathbf{\sigma}\xi^s(a^{r\dagger}_{-p}a^s_{-p}+a^{s\dagger}_{-p}a^r_{-p}) = -\int d^3p\xi^{rT}\mathbf{p}\mathbf{\sigma}\xi^sA(-p).$$ If this pesky ##A(-p)## can be shown to be equal to ##A(p)## or...

The above theorem is trying to find the pdf of a transformed random variable, it attempts to do so by "first principles", starting by using the definition of cdf, I don't understand why they have a ##f_X(x)## in the integral wouldn't ##\int_{\{x:r(x)<y\}}r(X) dx## be the correct integral for the...

I was very surprised to read the following in Needham, Visual Complex Analysis: "It is therefore doubly puzzling that the Trapezoidal formula is taught in every introductory calculus course, while it appears that the midpoint Riemann sum RM is seldom even mentioned." I was surprised because I...

I have done one by assuming tanx as u in substitution

Hi, PF, here goes an easy integral, meant to be an example of integration by parts. Use integration by parts to evaluate ##\int \sin^{-1}x \, dx## Let ##U=\sin^{-1}x,\quad{dV=dx}## Then ##dU=dx/\sqrt{1-x^2},\quad{V=x}## ##=x\sin^{-1}x-\int \frac{x}{\sqrt{1-x^2} \, dx}## Let ##u=1-x^2##...

Hi, suppose you have a non-zero smooth vector field ##X## defined on a manifold (i.e. it does not vanish at any point on it). Can its integral curves cross at any point ? Thanks. Edit: I was thinking about the sphere where any smooth vector field must have at least one pole (i.e. at least a...

For ##c > 0## and ##0 \le x \le 1##, find the complex integral $$\int_{c - \infty i}^{c + \infty i} \frac{x^s}{s}\, ds$$

For ##x\in \mathbb{R}##, let $$A(x) = \frac{1}{2\pi}\, P.V. \int_{-\infty}^\infty e^{i(xy + \frac{y^3}{3})}\, dy$$ Show that the integral defining ##A(x)## exists and ##|A(x)| \le M(1 + |x|)^{-1/4}## for some numerical constant ##M##.

Hi, PF 1-The elementary integral is ##\displaystyle\int{\displaystyle\frac{1}{a^2+x^2}dx}=\displaystyle\frac{1}{a}\tan^{-1}\displaystyle\frac{x}{a}+C## 2-The example is...

Hi, unfortunately I have several problems with the following task: I have problems with the tasks a, d and e Unfortunately, the Green function and solving differential equations with the Green function is completely new to me In task b, I got the following for ##f_h(t)=e^{-at}##.Task a...

Method 1, Pretty straightforward, $$\int_{-1}^0 |4t+2| dt$$ Let ##u=4t+2## ##du=4 dt## on substitution, $$\frac{1}{4}\int_{-2}^2 |u| du=\frac{1}{4}\int_{-2}^0 (-u) du+\frac{1}{4}\int_{0}^2 u du=\frac{1}{4}[2+2]=1$$ Now on method 2, $$\int_{-1}^0 |4t+2| dt=\int_{-1}^{-0.5} |4t+2|...

Hello, Please see this part of the article. I need to obtain the ##\rho (\phi)## value after obtaining the c0 to c5 constants of the ##\sigma (\phi)##. But as you can see after finding the coefficients, solving Eq.(1) could be a demanding job(I wasn't able to calculate the integral of Eq(1)...

TL;DR Summary: How to find integrals of parent functions without any horizontal/vertical shift? Say you were given the equation : How would you find : with a calculator that can only add, subtract, multiply, divide Is there a general formula?

I'm given the wavefunction and I need to find the normalization constant A. I believe that means to solve the integral The question does give some standard results for the Gaussian function, also multiplied by x to some different powers in the integrand, but I can't seem to get it into...

Using integration by parts: $$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$ $$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$ Then how to continue? Thanks

I encounter a function that I don‘t know in the calculation of Relativistically invariant 2-body phase space integral: in this equation, ##s##is the square of total energy of the system in the center-of-mass frame(I think) I don't know what the function ##\lambda^{\frac{1}{2}}## is. There are...

Evaluate the integral $$\int_0^1 x\left\{\frac{1}{x}\right\}\, dx$$ where ##\{\frac{1}{x}\}## denotes the fractional part of ##1/x##.

I want to find the cumulative mass m(r) of a mass disk. I have the mass density in terms of r, it is an exponential function: ρ(r)=ρ0*e^(-r/h) A double integral in polar coordinates should do, but im not sure about the solution I get.

The first step seems easy: computation of the $\theta$ and $\overline{\theta}$ integrals give $$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \det(\partial_j w_i(x)) \exp{\left(-\frac{1}{2}w_i(x)w_i(x)\right)}.$$ From here, I tried using that $$\det(\partial_j w_i (x)) = \det\left(\partial_j w_i...

I have the following problem and am almost sure of the answer but can't quite prove it: ##f(y)## is nonnegative, and I know that ##\int_0^{\infty } f(y) \, dy## is finite. I now need to calculate (or simplify) the double integral: $$\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \...