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If I have the coordinates of the points on the surface of the $n$-ball then, how can I find the volume? or how can I generalize trapezoidal rule for integration?

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The volume of an $n$-ball is given by $\frac{pi^\frac{n}{2}{\Gamma (\frac{n}{2}+1)}r^n$. Is this what you're looking for? – hombre Jan 12 at 18:45

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An easy way to do it would be using Monte Carlo integration. Surround the ball by a box of known volume and sample points inside the box, checking whether they're inside the ball.

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