A basic property of definite integrals is their invariance under translation, as expressed by the equation

The equation holds whenever ƒ is integrable and defined for the necessary values of x. For example in the accompanying figure, show that

because the areas of the shaded regions are congruent.

Use a substitution to verify Equation (1).

Transcribed Image Text:

·b [ - f(x) dx = a b-c a-c f(x + c) dx. (1)