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The capacity is equal to the integral of |gradient|$^2$ of spherically symmetric harmonic function with 1 on the bry of the ball and zero at infinity. The function $f$ depends only on the radius, say $r$. You can cook an ODE for $f$, something like $$f''(r)+\frac{(n-1){\cdot}f'(r)}{\sinh r}=0.$$ $f''(r)+\frac{(n-1){\cdot}\cosh r}{\sinh r}{\cdot}f'(r)=0.$$Then you should solve it and integrate$$\mathrm{vol}\, S^{n-1}{\cdot}\int\limits_R^\infty(f')^2{\cdot}(\sinh r)^{n-1}\, dr.$$(I do not know if will get the answer in a simple form, but it will be good for all practical purposes...) 2 deleted 4 characters in body Sounds like a home work; here are some hints: The capacity is equal to the integral of |gradient|^2 of spherically symmetric harmonic function with 1 on the bry of the ball and zero at infinity. The function f depends only on the radius, say r. You can cook an ODE for f, something like$$f''(r)+\frac{(n-1){\cdot}(f'(r))^2}{\sinh$f''(r)+\frac{(n-1){\cdot}f'(r)}{\sinh r}=0.$$Then you should solve it and integrate$$\mathrm{vol}\, S^{n-1}{\cdot}\int\limits_R^\infty(f')^2{\cdot}(\sinh r)^{n-1}\, dr.$$(I do not know if will get the answer in a simple form, but it will be good for all practical purposes...) 1 Sounds like a home work; here are some hints: The capacity is equal to the integral of |gradient|^2 of spherically symmetric harmonic function with 1 on the bry of the ball and zero at infinity. The function f depends only on the radius, say r. You can cook an ODE for f, something like$$f''(r)+\frac{(n-1){\cdot}(f'(r))^2}{\sinh r}=0.$$Then you should solve it and integrate$$\mathrm{vol}\, S^{n-1}{\cdot}\int\limits_R^\infty(f')^2{\cdot}(\sinh r)^{n-1}\, dr.