Let $v=(v_1,...,v_d)$ be a vector in $\mathbb{R}^d$ and consider the aligned rectangle $R=[a_1,b_1]\times...\times[a_d,b_d]$ in $\mathbb{R}^d$. Calculate the integral $\int_{R}^{}e^{-\langle x,v\rangle} dx$.
I approached the problem from an entirely quantitative point of view as the question is asked in that way, but I also want to know the qualitative arguments behind my calculations. Here is my approach and result:
$\int_{R}^{}e^{-\langle x,v\rangle} dx = \int_{a_d}^{b_d}...\int_{a_1}^{b_1}e^{-(x_1v_1+...+x_dv_d)}dx_1...dx_d$. At this point, using Fubini's Theorem and the fact that each function is a function of a single variable, I split the integrals according to the coordinate on which the integral is taken, which gives: $= \int_{a_d}^{b_d} e^{-(x_dv_d)}dx_d ... \int_{a_1}^{b_1} e^{-(x_1v_1)}dx_1 $, where for each $i=1,...,d$ we have $\int_{a_i}^{b_i} e^{-(x_iv_i)}dx_i = -\frac{1}{v_i}(e^{-b_iv_i} - e^{-a_iv_i}) $. Using this line of thinking which I am not sure, the result is: $\prod_{i=1}^{d} -\frac{1}{v_i}(e^{-b_iv_i} - e^{-a_iv_i}) $.
The part I have doubt is that when I simply split the integrals into $d$-many integrals on the real line: How should I support this transition with a rigorous approach? Is "by Fubini's Theorem" enough?