Integrate indicator function by Lebesgue measure

Question

So I'm struggling with the following integral: $\int_{[-1,1]^d} \mathbb{1_{||x||^2 \leq1}} d\lambda(x)$.

I know it's supposed to be the volume of the $d$-dimensional unit sphere, but I've trouble writing it down exactly. This is the result I arrived at, after calculating the expectation of a uniformly distributed random variable, transformed by a function containing among others the indicator left in the integral above.

Thanks ahead.

Answer 1

I think the integral is just equal to $\int...=\lambda(\{x\in [-1,1]^d\})= \frac{\pi^{d/2}}{\Gamma(\frac{d}{2}+1)} $, if we assume euclidean norm. Then the indicator can be ignored, as it doesn't change the domain of x.