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Given a vector (mathbf {a}in mathbb {R}^n), we provide an alternative and direct proof for the formula of the volume of sections (Delta cap {mathbf {x}: mathbf {a}^Tmathbf {x}u003c= t}) and slices (Delta cap {mathbf {x}:mathbf {a}^Tmathbf {x}= t}), (tin mathbb {R}), of the simplex (Delta ). For slices the formula has already been derived but as a by-product of the construction of univariate B-Splines. One goal of the paper is to also show how simple and powerful the Laplace transform technique can be to derive closed form expressions for some multivariate integrals. It also complements some previous results obtained for the hypercube ([0,1]^n).

Volume of slices and sections of the simplex in closed form