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If $f$ is continuous on $\mathbb{R},$ prove that $$\int_{a}^{b} f(x+c) d x=\int_{a+c}^{b+c} f(x) d x$$ For the case where $f(x) \geqslant 0,$ draw a diagram to interpret this equation geometrically as an equality of areas.

$\int_{a}^{b} f(x+c) d x=\int_{a+c}^{b+c} f(x) d x$

Calculus 1 / AB

Calculus 2 / BC

Chapter 5

Integrals

Section 4

The Substitution Rule

Integration Techniques

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Okay, so it says if efforts continues on our proof that the into go from a to B f f f x plus e T x is equal to the issue of a plus C to be fussy of f of x t act. So to prove this, we're gonna have f of x equals X and X plus c We're going to find season with one, So we have X plus one. So therefore, you can quite clearly see that a one is equal to eight years. So even if you just, like, take a mental snapshot of this area here and then bring it over this area here, you're gonna see that they are in fact equal.

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