Prove an equivalent form of the translation property

by RoRi

July 19, 2015

The translation property (Theorem 1.7) states

$\int_a^b s(x) \, dt = \int_{a+c}^{b+c} s(x-c) \, dx \qquad \text{for all } c \in \mathbb{R}.$

Prove that the following is equivalent to the translation property:

$\int_{a+c}^{b+c} f(x) \, dx = \int_a^b (x+c) \, dx.$

Proof. Let $d = a+c$ and $e= b+c$ . Then, by the translation property we have:

$\int_d^e f(x) \, dx = \int_{d-c}^{e-c} f(x-(-c)) \, dx$

using $-c \in \mathbb{R}$ in the theorem. Substituting $d = a+c$ and $e=b+c$ back into the equation, we obtain,

$\int_{a+c}^{b+c} f(x) \, dx = \int_a^b f(x+c) \, dx. \qquad \blacksquare$

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