Chapter 5.5, Problem 114E

Substitution: shift Perhaps the simplest change of variables is the shift or translation given by u = x + c, where c is a real number.

1. a. Prove that shifting a function does not change the net area under the curve, in the sense that

${\displaystyle {\int }_a^{b}f(x+c)}dx={\displaystyle {\int }_{a+c}^{b+c}f(u)du}.$

1. b. Draw a picture to illustrate this change of variables in the case that f(x) = sin x, a = 0, b = π, and c = π/2.