How to evaluate the following integral in Matlab? $\int_a^b{\int_0^{2\pi}{\sqrt{y^2(\cos^2{\theta}-1)+1}}\,\mathrm{d}\theta}\,\mathrm{d}y$
$a$ and $b$ are vectors with values.
How to evaluate the following integral in Matlab? $\int_a^b{\int_0^{2\pi}{\sqrt{y^2(\cos^2{\theta}-1)+1}}\,\mathrm{d}\theta}\,\mathrm{d}y$
$a$ and $b$ are vectors with values.
Use the dblquad function.
dblquad(@(t,y)sqrt(y.^2*(cos(t).^2-1)+1), 0, 2*pi, a, b)
I can't tell what evaluating an integral with vector limits means, unless you're doing things componentwise. In that case, you're asking how to evaluate
$\int_a^b{\int_0^{2\pi}{\sqrt{1-y^2\sin^2{\theta}}}\,\mathrm{d}\theta}\,\mathrm{d}y$
which simplifies by symmetry to
$4\int_a^b{\int_0^{\pi/2}{\sqrt{1-y^2\sin^2{\theta}}}\,\mathrm{d}\theta}\,\mathrm{d}y$
and the complete elliptic integral of the second kind pops up:
$4\int_a^b{E(y^2)}\,\mathrm{d}y$
where $E(m)$ is the complete elliptic integral of the second kind with parameter $m$, as implemented in MATLAB.
You can then use the usual quadrature routines (e.g. quad()
) like so: construct the appropriate function
function e = osama(x) [k, e] = ellipke(x^2);
and then feed the function handle to the quadrature routine: 4.*quad(@osama,a,b)
.
As for a closed form, it involves hypergeometric functions, so I doubt such a thing would be of any use to you.