Integral Definition and 1000 Threads

Dear Every One, I have a question: What does an integral multiple of 4 means?Thanks Cbarker1

Homework Statement This is not a homework question, just a general wonderment , how can I integrate the following wrt time? Homework Equations \dot{\textbf{r}}.\ddot{\textbf{r}} +G(m_1 + m_2)\frac{\dot{\textbf{r}}}{r^2} = 0 The Attempt at a Solution The solution is given as \frac{1}{2}v^2 -...

Hey! :o Using polar coordinates I want to calculate $\iint_D \frac{1}{(x^2+y^2)^2}dxdy$, where $D$ is the space that is determined by the inequalities $x+y\geq 1$ and $x^2+y^2\leq 1$. We consider the function $T$ with $(x,y)=T(r,\theta)=(r\cos \theta, r\sin\theta)$. From the inequality...

Consider the integral $$ G(x) = \int_0^\infty \frac{e^{-xt}}{1+t}dt$$ which is convergent for x>0. For large x, it is dominated by small t so expand: $$G(x) = \int_0^\infty e^{-xt}\sum_{m=0}^{\infty}(-t)^mdt$$ From here my notes say to take out the summation and write: $$G(x) =...

Hi. I know how to use Gauss' Law to find the electric field in- and outside a homogeneously charged sphere. But say I wanted to compute this directly via integration, how would I evaluate the integral...

Homework Statement The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems. For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...

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 $x^4 + 8y^4+y = 1$.

15.3.65 Improper integral arise in polar coordinates $\textsf{Improper integral arise in polar coordinates when the radial coordinate r becomes arbitrarily large.}$ $\textsf{Under certain conditions, these integrals are treated in the usual way shown below.}$ \begin{align*}\displaystyle...

$\displaystyle \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \sqrt{x^2+y^2} \, dydx=\frac{\pi}{6}$ 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

Consider the Kirkoff integral theorem and the Huygens -Fresnel principle/formula (both from Wikipedia): KIT The Kirchoff integral for monochromatic wave is: $$U({\mathbf {r}})={\frac {1}{4\pi }}\int _{S'}\left[U{\frac {\partial }{\partial {\hat {{\mathbf {n}}}}}}\left({\frac...

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

$\tiny{232.q1.5,a}$ \begin{align*}\displaystyle I_a&=\iint\limits_{R} xy\sqrt{x^2+y^2} \, dA \\ R&=[0,2]\times[-1,1] \end{align*} would this be $$\int_{-1}^{1} \int_{0}^{2}xy\sqrt{x^2+y^2} \,dx \, \, dy $$

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*}

I have a (somewhat) strange energy equation which has the following form: KE = A + B W + C \exp(-D W), where A,B,D are known constant, C is an unknown constant to be determined and kinetic and potential energy are given by KE and W respectively with W\equiv W(r) i.e. is a function of...

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)...