Questions tagged [elliptic-integrals]

Questions on elliptic integrals, integrals that involve the square root of a cubic or quartic polynomial.

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Assistance with solving the integral

Can you give me an idea how to handle this integral? $\int_{0}^{2\pi} (1+\cos{x})\sqrt{3+\cos{x}}\,dx$
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2 votes
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Elliptic integrals of complex argument and parameter

Mathematica has the annoying habit to provide solutions involving incomplete elliptic integrals of the first kind $E(z|m)$, second kind $F(z|m)$ and third kind $\Pi(...
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Elliptic integrals and average distance from the origin

Consider any linear map $\mathbf{R}^2 \to \mathbf{R}^2$, represented as a matrix $M$. If $B$ is the unit disc, then $MB$ is an ellipse, which we can assume to have at its axes on the coordinate ...
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4 votes
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Any hint for evaluating $\int_0^\theta \sec^2(\phi)\cdot\sqrt{\sec2\phi}\,d\phi\;$?

How can I evaluate this integral? $$\int_0^\theta \sec^2(\phi)\cdot\sqrt{\sec2\phi}\,d\phi$$ I tried integral by parts using $u = \sqrt{\sec2\phi}$ and $dv =\sec^2(\phi) d\phi$ but ended up falling ...
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Simplifications of the Elliptic integral $\displaystyle J_n(r)=\int^r_{r_0}\frac{x^n}{\sqrt{(x-x_1)(x-x_2)(x-x_3)}}$ for specific values of $n$.

Consider an indefinite "elliptic integral”, $$ J_n(r):=\int^r_{r_0}\frac{x^n}{\sqrt{f(x)}}, \quad f(x):=(x-x_1)(x-x_2)(x-x_3). $$ where $r_0>0, n=1,0,−1$, and $x_{1,2,3}$ are roots of the ...
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-2 votes
2 answers
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prove the definite Integral

Hey is anyone here who can prove that: $$\int_0^{\pi/2} \sqrt{2\sin(x)} dx = \frac{2\Gamma(\frac{3}{4})^2}{\sqrt{\pi}}$$ we haven't had the Gamma Function in our lectures. But i just want prove that ...
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1 vote
1 answer
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What is the argument of the quarter period $iK^\prime(k)$ when $|k| = 1$, i.e. $k = e^{i\alpha}$? (Is it $\frac{\pi-\alpha}{2}$?)

I have a research problem in mathematical physics describing a periodic condensate. I've determined a class of solutions wherein the density of the condensate is described by an elliptic function, ...
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-1 votes
2 answers
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how to prove this identity $s = 2 \int_{- \pi}^{\pi} | \frac{\sin (t) - i}{(\sin (t) + i)^2} | dt = 2 K (- 1) = 2.62206...$

How can we prove this identity? Which, btw, Mathematica know how to simplify so it is missing some fundamental identity (related to the lemniscate constant.) \begin{equation} s = 2 \int_{- \pi}^{\pi}...
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3 answers
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Integral of $\sqrt{\cosh(x)}$ with respect to x

I am trying to obtain a solution for the integral \begin{equation} \int^{x}_{0} \sqrt{\cosh(x)} dx. \end{equation} A CAS system yields an answer depending on an elliptic integral of the second kind ...
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2 votes
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Closed form of $\int \frac{1}{\sqrt{(x-a) (x-b) (x-c) (x-d) (x-e)}} \, \rm{d}x$

I want to find the closed form of this integral: $$ I_5=\int \frac{1}{\sqrt{(x-a) (x-b) (x-c) (x-d) (x-e)}} \, \mathrm{d}x $$ where $a<b<c<d<e$. My attempt My idea is to find a ...
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Prove that $\int_{0}^{1} x K(x) E(x) \,\mathrm{d} x{=}\frac{9}{16}+\frac{21}{32} \zeta(3)$

Background let $\displaystyle I_k=\int_{0}^{1} x^k K(x) E(x) \,\mathrm{d} x$. where $K(z)$ is the complete elliptic integral of the 1st kind, $E(z)$ is the complete elliptic integral of the 2nd kind. ...
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1 vote
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Defnite integral of the square root of a fourth degree polynomiae

I am trying to solve $$ \int_{b_1}^{b_2}\sqrt{(b-b_1)(b_2-b)(b_3-b)(b_4-b)} db $$ where $ b1 < b2 < b3 < b4 $. I am quite sure this is some linear combination of elliptic integrals, but I ...
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4 votes
1 answer
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Does the Incomplete Beta function have forms of Elliptic E besides $\frac14 \text B_{\sin^2(2x)}\left(\frac12,\frac34\right)=\text E(x,2)$?

Goal: To find more special cases of the Incomplete Beta function $\text B_z(a,b)$ in terms of Elliptic $\text E(x,k)$ using Mathematica notation: The goal is to find values of: $$\text B_z(a,b)=\int z^...
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2 votes
1 answer
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Show that $\int_0^{\pi/2} \frac{dt}{\sqrt{\sin t}}=\int_0^1 \frac{dx}{\sqrt{x-x^3}}$

This is intended to be an example of an elliptic integral but I'm not sure how to go about showing it. I'm not sure which identities to start with, any tips would be greatly appreciated!
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0 votes
2 answers
137 views

Is it possible to compute $\int_0^{\pi/2}\sqrt{a^2 \sin^2 x + b^2 \cos^2 x}\ dx$. [closed]

Basically, this integral, except the $\ln$ is replaced with $\sqrt\cdot$. In a question we were asked to find the circumference of part of an ellipse, and we got this integral, so I was wondering if ...
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2 votes
0 answers
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Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
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6 votes
1 answer
128 views

Closed form for definite integrals invovling Jacobi elliptic functions

In a 1879 work, Glaisher proves the following closed forms $$\int_{0}^{K\left(k\right)}\log\left(\text{sn}\left(z;k\right)\right)dz=-\frac{1}{4}\pi K^{\prime}\left(k\right)-\frac{1}{2}K\left(k\right)\...
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0 votes
0 answers
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Solving $ \int \sqrt{4\sin^2(x) + \cos^2(x) } dx $ without an elliptical integral

Solving: $$ \int \sqrt{4\sin^2(x) + \cos^2(x) } dx $$ I have used calculators attempt this problem, but they all use the elliptical integral of the second parameter. My goal is to somehow solve this ...
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5 votes
2 answers
208 views

Computing : $\displaystyle \int \frac {m + n\cos (x-x_0) }{ [a+b\cos (x-x_0)]^{3/2} } \mathrm{d}x $ with integration of $E$ and $F$

Yesterday I tried to find an expression to predict the behaviour of Magnetic Field due a circular loop at any point in the space, in cylindrical co-ordinates. At the end, I am stuck with some ...
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0 answers
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Generalization of Elliptic Integrals?

Elliptic integrals calculate the arclength of an ellipse. But, a 3-dimensional ellipsoid has three diameters. Is there an integral formula that generalizes the arclength of an ellipsoid surface in 3 ...
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4 votes
1 answer
133 views

Proving $\frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}\big(P_{2l}(x)+\text{sgn}(x)P_{2l+1}(x)\big)$

Can someone help me in proving the following: $$ \frac{\pi}{2}=\sum^\infty_{l=0} \frac{(-1)^l}{2l+1}(P_{2l}(x)+\text{sgn}(x)\cdot P_{2l+1}(x)), $$ for any value of $x$, $-1\le x\le 1$? (Here $P_l(x)$ ...
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1 vote
0 answers
112 views

Evaluate $\int_{0}^{1}K(x)^2\text{d}x -\int_{0}^{1} \frac{x\sqrt{1-x^2} }{2-x^2}K(x)^2\text{d}x$

Recently, I found this identity on a mathematical site(seems true): $$\int_{0}^{1}K(x)^2\text{d}x -\int_{0}^{1} \frac{x\sqrt{1-x^2} }{2-x^2}K(x)^2\text{d}x =\frac{\Gamma\left ( \frac{1}{4} \right )^...
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5 votes
0 answers
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Is this function an alternative solution to the nonlinear pendulum?

Is this function an alternative solution to the nonlinear pendulum? Introduction I am working with the differential equation of the frictionless nonlinear pendulum: $$\ddot{\theta}(t) + b\,\sin(\theta(...
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1 vote
1 answer
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Hyperelliptic integral: Piecewise solution (handbook) vs General solution (Wolfram)

Recently, I posted a question asking for hints on how to solve the integral $$\int \frac{{\rm d}x}{-\sqrt{x^6+x^2+a}}$$ for which I received an amazing answer. The answer helped me reduce the integral ...
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6 votes
0 answers
116 views

Evaluation of $\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt[3]{\frac{1}{3} + \sin^2 x}} $ [duplicate]

I want to evaluate the following integral in a closed form involving elementary functions: $$I = \displaystyle \int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt[3]{\frac{1}{3} + \sin^2 x}} = \frac{\sqrt[3]{3}\...
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0 votes
1 answer
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Is there an easy way to convert a partial arc length or movement along an elliptic curve into an X,Y position?

I had this idea to project the Earth onto a 2D map using elliptical cylinders that could be unrolled. (The Earth can be well approximated by rotating an ellipse on its axis to create an oblate ...
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3 votes
1 answer
138 views

Hyperelliptic integral involving square root of sextic polynomial

I am interesting in finding the integral of $$ I = \int \frac{{\rm d}x}{-\sqrt{x^6+x^2-c}}$$ Where $c \leq x^6 + x^2$ such that the integral is always real. I was trying to look at a reduction of 579....
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  • 144
4 votes
1 answer
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Proving $\int_{0}^{1} \frac{K(x)K\left ( \sqrt{2} \sqrt{(1-x^2)/(2-x^2)} \right ) }{2-x^2}\text{d}x =\frac{\pi^3}{8\sqrt{2} } {}_6F_5(...)$

I encountered an integral identity: $$\int_{0}^{1} \frac{K(x)K\left ( \sqrt{2} \sqrt{\frac{1-x^2}{2-x^2} } \right ) }{2-x^2}\,dx =\frac{\pi^3}{8\sqrt{2}} {}_6F_5\left ( \frac{1}{4},\frac{1}{4},\...
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0 votes
1 answer
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Separable ODE - Integral involving Elliptic integral

I am trying to solve the equation $$ \dot y(t) = -\sqrt{ y(t)^6 + y(t)^2 + a } $$ with $a \leq y(t)^6+y(t)^2$ for all $t\geq 0$. This is a first order separable ODE. According to Wolfram Alpha the ...
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8 votes
0 answers
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An Elliptic Curve has 3 real roots if and only if $\frac{\omega_2}{\omega_1}$ is purely imaginary

Define $\Lambda=\{n\omega_1+m\omega_2 \mid n,m\in \mathbb{Z}\}$ for $\omega_1, \omega_2\in \mathbb{C}$ and $\wp(z)$ is the corresponding Weierstrass elliptic function. I want to show that $e_1=\wp(\...
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0 votes
0 answers
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Legendre's formula for the surface area of scalene ellipsoid

I am looking for a derivation of the formula $$S=2\pi c^2 +\frac{2\pi a b}{\sin \varphi}\left(E(\varphi, k)\sin ^2 \varphi +F(\varphi,k)\cos^2 \varphi\right)$$ for the surface area of the ellipsoid $$\...
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  • 328
2 votes
1 answer
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Is this Lauricella $\text F_\text D$ to hypergeometric R, from DLMF, conversion formula correct?

Here is another post about uncommon standard special functions in the form of Elliptic Integrals which will be special cases of the Lauricella D function. Most elliptic integrals can be put in terms ...
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0 votes
2 answers
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Can the complete elliptic integrals of the third kind to be expressed in series?

Can the complete elliptic integrals of the third kind which are defined by $$ \Pi (\eta,κ)=\int_0^{\pi/2} d\theta \frac {1}{\sqrt{1−κ \sin^2\theta}} \frac{1}{1 -ηsin^2\theta }$$ to be expressed in ...
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0 answers
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Solving integrals of functions of Jacobi elliptic amplitudes

I have an integral on the form $\int \operatorname{dn}(\frac{x}{k},k^2)^2 + \sin({N\pi/2} + \operatorname{am}(\frac{x}{k}, k^2))^2dx$, anyone know of any good sources or solution methods for suchs an ...
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1 vote
0 answers
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Is the solution to the Weierstrass ODE over the reals still the Weierstrass elliptic function?

Suppose that we have a function $f(x)$ of a real variable $x$ in some domain of $U\subseteq\mathbb R$. Also, suppose that this function satisfies the ODE $$(f'(x))^2=4f^3(x)-g_2f(x)-g_3\,,$$ which is ...
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3 votes
0 answers
121 views

A "mysterious" sequence?

A recent question asked for the Taylor expansion of $$f(x)=\exp\left(-\pi\,\frac {_2F_1(\frac12, \frac12, 1; 1-x)}{_2F_1(\frac12, \frac12;1;x)}\right)=\exp\left(-\pi\,\frac{K(1-x)}{K(x)}\right)=\frac{...
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0 answers
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Question about addition of points on elliptic curves in relation to elliptic integrals

I'm working on a long form piece on the development of elliptic curve cryptography, basically tracing the use of a point at infinity back to the discovery of the rules for linear perspective in the ...
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3 votes
1 answer
69 views

Numerical Integral of Complete Elliptic Integral

I'm looking to numerically evaluate an integral of the form $$I=\int_0^1rK(r)f(r)\,dr$$ where $f(r)$ is a smooth function known at a set of grid points $\{r_i\}$ with error $O(\Delta r^2)$ ($\Delta r\...
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  • 777
1 vote
1 answer
88 views

On integration involving elliptic integral

I'm working on the following integral involving an elliptic integral. $E(m)$ is the complete elliptic integral of second kind with parameter $m=k^2$. Is there a way to solve this? Or, is it impossible ...
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1 vote
1 answer
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Legendre relations for elliptic integrals with null imaginary part

I am trying to compute the transformation of $K(k)$ as $k \to 1/k$, where $K(k)$ is just the Legendre integral $$K(k) := F\left(\frac{\pi}{2}, k\right).$$ The Digital Library of Mathematical Functions ...
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  • 151
4 votes
0 answers
36 views

An integral including the complete elliptic integral

Let $\lambda>1$, $0<\alpha<\pi$, how to evaluate: $$I=\int_0^\pi K\left(\sqrt{\frac{2\sin\theta\sin\alpha}{\lambda-cos(\theta+\alpha)}}\right)\frac{\sin\theta\cos\theta d\theta}{\sqrt{\lambda-...
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  • 301
0 votes
1 answer
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Arithmetic mean geometric mean 2d map

Consider the 2d iteration \begin{aligned} x_{n+1} &=\frac{x_n+y_n}2,\\ y_{n+1} &=\sqrt{x_ny_n}. \end{aligned} We assume that $x_0>y_0>0$. Hence, $x_n>x_{n+1}>y_{n+1}>y_n$...
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  • 817
0 votes
0 answers
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Integration of Volume of Intersection of Cylinders

Volume of Intersection of cylinders (different radii) Integrating a specific integral based on the volume of the intersection of two cylinders These links above describe how to integrate the volume of ...
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  • 31
4 votes
0 answers
110 views

Evaluate the elliptic integral $ \int_{0}^{\infty} \frac{dx}{\sqrt{112+21x^2+x^4}}$ [duplicate]

I found the following integral and I am stuck with the procedure to find the given closed form in terms of the complete Gamma function $$\displaystyle I= \int_{0}^{\infty} \frac{dx}{\sqrt{112+21x^2+x^...
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0 votes
2 answers
94 views

Convert elliptic: $F\left(\sin^{-1}\left(\sqrt{\frac{R}{R+2}}\right)\biggr\rvert 1-\frac{4}{R^2}\right)$ to complete: $K\left(1-\frac{4}{R^2}\right)$

I explored the convolution of $\arcsin$ distributions SE. Here I found the following identity empirical (allot of trial, error and luck): typo: $1-\frac{1}{R^2}$ should be $1-\frac{\color{red}{4}}{R^...
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0 votes
0 answers
67 views

Connection between elliptic integrals and theta functions

I have read about connections between elliptic integrals and their connections to the Jacobi theta functions, like $\theta_3^2(q) = \frac{2}{\pi}K(k)$, where $q=e^{-\pi\frac{K’(k)}{K(k)}}$, but how ...
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1 vote
0 answers
90 views

Integrating the square root of a trigonometric polynomial [duplicate]

I am trying to find a solution to the following integral. $$ \int \sqrt {17+8\cos(3t)+9\cos^2(3t)} dt$$ I have run into exactly the same problem as this previous unanswered question. This integral ...
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2 votes
1 answer
36 views

How to show Elliptic integral equality?

I am trying to prove the complex period of Jacobi elliptic functions are $2K + 2iK’$ and $4K + 4iK’$. The crucial step that I am missing is to show the following equality: $$\int_{0}^{1} \frac{\mathrm{...
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2 votes
2 answers
70 views

Find the arc length $\int_ 0 ^{4\pi} r(t)=3\cos ti+4\sin tkj+tk$

So my components are $$f(x)= 3\cos t, \ \text g(y)=4\sin t ,\ \text h(z)= t$$ $$f'(x)=-3\sin t, \ \text g'(y)=4\cos t,\ \text h'(z)=1$$ I found the derivatives of each component and plugin to the arc ...
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2 votes
0 answers
91 views

Definite integral with square roots

I want to compute the following definite integral, where $z\in(0,1)$: $$ \int_{-1}^{1} dx \int_{-1}^{1} dy \ \frac{z^2 \sqrt{1-x^2} \sqrt{1-y^2} \sqrt{z+\frac{1}{z}-2 x} \sqrt{z+\frac{1}{z}-2 y}}{\pi ^...
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