Integral Definition and 1000 Threads

Write interated integrals in spherical coordinates for the following region in the orders $dp \, d\theta \, d\phi$ and $d\theta \, dp \, d\phi$ Sketch the region of integration. Assume that $f$ is continuous on the region \begin{align*}\displaystyle...

I was wandering if there is a way to understand whether it is possible to find an indefinite integral of a function. Let's say e^(-x^2) or e^(x^2)... They can't have indefinite integrals, but how can I say it? Is there a theorem or something?

Evaluate the following: $$I=\int_0^{\infty} xe^{ax}\cos(x)\,dx$$ where $a<0$

So My Question Is Simple, But It confuse me too much! What Is Difference between the notation ∫d3x and ∫∫∫dxdydz ?

Evaluate the definite integral:\[I = \int_{0}^{\pi}\frac{\cos 4x - \cos 4\alpha }{\cos x - \cos \alpha }dx\]- for some $\alpha \in \mathbb{R}.$

Homework Statement Solve from x = 0 to x = ∞, ∫xe-axcos(x)dx Homework EquationsThe Attempt at a Solution I have a solution for the integral ∫e-axcos(x)dx at the same limits, and I feel that the result might be of use, but have no idea how to manipulate the integral above such that I can use...

What is the justification for differentiating some integrals with respect to constants in order to obtain result, i.e. ∂/∂a[∫e^(− ax^2).dx] =∫-x^2.e^(-ax^2) dx?I mean what if we say "a" was 3 then differentiating wrt 3 would have no significance?How can we treat it like a multivariable function :/

$\tiny{232.15.4.46}$ $\textsf{Change the order then evaluate}$ \begin{align*}\displaystyle I&=\int_{0}^{1}\int_{0}^{2}\int_{2y}^{4} \frac{5\cos(x^2)}{2z} \, dx \, dy \, dz \end{align*} ok I presume the change that should be made is... altho I don't know what represents x or y...

calculate the line integral for a vector field F= -xy⋅j over a circle which is c(t)=costi+sintj, so I used x=cost y=sint and ∫(0 to 2pi) -(sintcost)(cost)dt=(cos^3(2pi)-cos^3(o))/3=0 now here is the problem, if this enclosed line integral is zero then why is the vector field not conservative?

Hi, I'm stuck on a problem from my quantum homework. I have to show <p1|p2> = ∫(from -1 to 1) dx (p1*)(p2) is a scalar product (p1 and p2 are single variable complex polynomials). I've figured out how to show that they satisfy linearity and positive definiteness, but I'm completely stuck on...

\begin{align*}\displaystyle \int_{\alpha}^{\beta}\int_{a}^{\infty} g(r,\theta) \, rdr\theta =\lim_{b \to \infty} \int_{\alpha}^{\beta}\int_{a}^{b}g(r,\theta)rdrd\theta \end{align*} $\textit{Evaluate the Given}$ \begin{align*}\displaystyle &=\iint\limits_{R} e^{-x^2-y^2} \, dA \\ (r,\theta) \, 2...

So when finding the Area from a double integral; or Volume from a triple integral: If the curve/surface has a negative region: (for areas, under the x axis), (for volumes, below z = 0 where z is negative) What circumstances allow the negative regions to be taken into account as positive when...

Evaluate the double integral: \[I = \int \int _R\frac{1}{(1+x^2)y}dxdy\] - where $R$ is the region in the upper half plane between the two curves: $2x^4+y^4+ y = 2$ and $x^4 + 8y^4+y = 1$.

$\displaystyle \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \sqrt{x^2+y^2} \, dydx=\frac{\pi}{6}$ this was the W|A answer but how ? also supposed to graph this but didn't know the input for desmos

Homework Statement I need find the function ##F(x)## . Homework Equations ##\int_0^r F(x)dx = \frac{r^3}{(r^2+A)^{3/2}}+N## where ##A,N## are constants. The Attempt at a Solution I tried using some function of test, for instance the derivative of the right function evaluated in x. But , i...

$\displaystyle \int_{-1}^{1} \int_{-2}^{3}(1-|x|) \,dy\,dx$ ok i was ? about the abs

$$\iint_\limits{R}(3x^5-y^2\sin{y}+5) \,dA$$ $$R=[(x,y)|x^2+y^2 \le 5]$$

Homework Statement An imperfect gas obeys the equation (p+\frac{a}{V^2_m})(V_m-b)=RT where a = 8*10^(-4)Nm^4mol^(-2) and b=3*10^(-5)m^3mol^(-1). Calculate the work required to compress 0.3 mol of this gas isothermally from a volume of 5*10^(-3)m^3 to 2*10^(-5)m^3 at 300K. Homework Equations...

Homework Statement question : find the value of \iint_D \frac{x}{(x^2 + y^2)}dxdy domain : 0≤x≤1,x2≤y≤x Homework Equations The Attempt at a Solution so here, i tried to draw it first and i got that the domain is region in first quadrant bounded by y=x2 and y=x and i decided to convert...

ok just seeing if I have this set up correctly before evaluate.. where does $15x^2$ come from? if $15x^2$ is inside this why would we need all the R values

Evaluate the integral: \[ \int_{0}^{\infty}\frac{\arctan(ax)-\arctan(bx)}{x}dx\] where $a,b \in \mathbb{R}_+$

Draw the regions of integration and write the following integrals as a single iterated integral. $$\displaystyle\int_{0}^{1} \int_{e^y}^{e} f(x,y)\,dx\,dy + \int_{-1}^{0} \int_{e^{-y}}^{e}f(x,y) \,dx\,dy$$ ok haven't done this before so kinda clueless

Use double integral to compute the area of the region bounded by $y=4+4\sin{x}$ and $y=4-4\sin{x}$ on the interval $\left[0,\pi\right]$ ok it looks easier to do this in one $\int$ but it asks for a double $\int\int$ so ?

ok so there are 3 peices to this Express and integral for finding the area of region bounded by: \begin{align*}\displaystyle y&=2\sqrt{x}\\ 3y&=x\\ y&=x-2 \end{align*}

If $$\phi(t,x)$$ is a solution to the one dimensional wave equation and if the initial conditions $$\phi(0,x) , \phi_t(0,x)$$ are given, D'Alembert's Formula gives $$\phi(t,x)= \frac 12[ \phi(0,x-ct)+ \phi(0,x+ct) ]+ \frac1{2c} \int_{x-ct}^{x+ct} \phi_t(0,y)dy . \tag{1}$$ which is...

$\textsf{a. Evaluate :}$ \begin{align*}\displaystyle R&=[0,2] \times [-1,1]\\ II_{5a}&=\iint\limits_{R}xy\sqrt{x^2+y^2}\, dA \end{align*} next step? $$\displaystyle\int_0^1 \int_{-1}^1 xy\sqrt{x^2+y^2}\, dxdy$$

$\textsf{d. Evaluate :}\\$ \begin{align*}\displaystyle II_{5d}&=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \frac{1}{(x^2+1)(y^2+1)} \, dy dx \end{align*}

Homework Statement Let ##f: (1, \infty) \to [0,\infty)## be a function such that the improper integral ##\int_{1}^{\infty} f(x)dx## converges. If ##f## is monotonically decreasing, then ##\lim_{x \to \infty} f(x)## exists. Homework EquationsThe Attempt at a Solution This problem doesn't come...

$\tiny 15.1.25$ $\textsf{Evaluate the following double integral over the region R}\\$ $\textit{note: the R actually is supposed be under both Integrals don't know the LaTEX for it}$ \begin{align*}\displaystyle \int_R\int&=5(x^5 - y^5)^2 dA\\ R&=[(x,y): 0 \le x \le 1, \, -1 \le y \le -1]...

Hello, I need to find the matrix elements of in the particular case where l = 1. This should have an analytical solution but I have no idea where to start with this demonstration. Any suggestions on where to start digging?Ty!

Calculate the integral: \[I = \int_{0}^{\frac{\pi}{4}}\left(\tan x + \cot x \right)\left ( \frac{\tan x}{1 + \cot x} \right )^2dx.\] A solution without the use of an online integral calculator is preferred. :cool:

Integral constant for internal energy of ionic liquid I have a question, and I will be really grateful if someone helps me. I have a polynomial equation for internal energy which I calculated by integration an equation of state formula, which is based on density. But, because I calculated this...

Integral constant for internal energy of ionic liquid I have a question, and I will be really grateful if someone helps me. I have a polynomial equation for internal energy which I calculated by integration an equation of state formula, which is based on density. But, because I calculated this...

Hi, I have this integral that I really want to calculate for a personal project (not for school), so I typed it into WolframAlpha and it said that the it took too long to compute and to get it computed I would have to pay money. Is there any free software that may be able to calculate this...

Hey I was just practicing Gauss's law outside a sphere of radius R with total charge q enclosed. So I know they easiest way to do this is: ∫E⋅da=Q/ε E*4π*r^2=q/ε E=q/(4*πε) in the r-hat direction But I am confusing about setting up the integral to get the same result I tried ∫ 0 to pi ∫0 to...

Homework Statement Hi! I need to find the limit when x-> +infinity of (integral from x to x^2 of (sqrt(t^3+1)dt))/x^5 Homework EquationsThe Attempt at a Solution The integral of (sqrt(t^3+1)dt) can only be estimated, so sqrt(t^3+1)=(t^(3/2))*sqrt(1+1/t^3) should I use the maclaurin series...

In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (page 17) they are discussing including an external field in a tight binding model, the Peierls...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ... I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the...

I note the following: \begin{equation} \begin{split} \langle \vec{x}| \hat{U}(t-t_0) | \vec{x}_0 \rangle&=\langle \vec{x}| e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} | \vec{x}_0 \rangle \\ &=e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} \delta(\vec{x}-\vec{x}_0)...

I have learned an adequate amount of calculus including implicit, parametric differentiation as well as Upton second order differential equations in high school math course. It was really abstract and we were taught only how to solve mathematical problems. Now, I need to model those problems in...

Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...

I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed. Homework Statement Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl## along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.Homework...

Calculate the following definite trigonometric integral: \[\int_{0}^{\frac{\pi}{2}} \cos^{2017}x \sin^{2017}x dx\].

$\tiny{242 .10.09.8}\\$ $\textsf{Express the integrand as a sum of partial fractions and evaluate integral}$ \begin{align*}\displaystyle I&=\int f \, dx = \int\frac{\sqrt{16+5x}}{x} \, dx \end{align*} \begin{align*}\displaystyle f&=\frac{\sqrt{16+5x}}{x}...

Hi, I have homework question that I'm trying to solve. But I can't understand the basis. Here is a picture of the question and what I have done: My question is, How do I set the boundaries for the integral? 1. If I want the whole squart. 2. To sum up areas. Assume I want to sum the areas, I...

Hello! I am reading a derivation of the path formulation of QM and I am a bit confused. They first find a formula for the propagation between 2 points for an infinitesimal time ##\epsilon##. Then, they take a time interval T (not infinitesimal) and define ##\epsilon=\frac{T}{n}##. Then they sum...

Charles Link submitted a new PF Insights post An Integral Result from Parseval's Theorem Continue reading the Original PF Insights Post.

Hi! I came across a proof in my physics textbook (amperage=wattage/area), and it contained this integration: ∫0 T sin2(ωt) dt The whole thing: 1/T∫0 T sin2(ωt) dt = 1/T(t/2 + sin2ωt/2ω)|T 0 = 1/2 I didn't remember how to integrate that, so I went back to check my notes, and look at it at...

Hi at all On my math methods book, i came across the following Fredholm integ eq with separable ker: 1) φ(x)-4∫sin^2xφ(t)dt = 2x-pi With integral ends(0,pi/2) I do not know how to proceed, for the solution...