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**ThePerfectHacker** Let $\displaystyle I = \smallint_a^b f$ we know that for any $\displaystyle \epsilon > 0$ there exists $\displaystyle \delta > 0$ so that $\displaystyle |\mathcal{R}(f,P) - I| < \epsilon$ where $\displaystyle P$ is any partition with $\displaystyle ||P||<\delta$ and $\displaystyle \mathcal{R}(f,P)$ is any associated Riemann sum. Let $\displaystyle P_k$ be any partition of $\displaystyle [a-k,b-k]$ with $\displaystyle ||P_k|| < \delta$. If $\displaystyle \sum_{j=1}^n g_k(t_j)(x_j-x_{j-1})$ is some Riemann sum then it is equal to $\displaystyle \sum_{j=1}^n f(x_j+k) [(x_j+k)-(x_{j-1}+k)]$, but this is a Riemann sum associated with $\displaystyle f$ where the partition norm is less than $\displaystyle \delta$. Thus, $\displaystyle \left| \sum_{j=1}^n g_k(t_j)(x_j-x_{j-1}) - I \right| < \epsilon$, so we see that $\displaystyle g_k$ is integrable with the same value.