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This is a prove question, I can't figure out how to do it, can anyone please help me? I would really appreciate it!


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marked as duplicate by Marvis, Gerry Myerson, Brett Frankel, Brandon Carter, Chris Eagle Feb 9 '13 at 23:29

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

possible duplicate of How is the Integral of $\int\_a^bf(x)~dx~~=\int\_a^bf(a+b-x)~dx$ This is a strict subset of the question, I have answered here. – user17762 Feb 9 '13 at 22:21
up vote 2 down vote accepted

To Prove: $$\int_0^a f(x)\;dx=\int_0^a f(a-x)\;dx\\$$ Proof:

\begin{align} RHS&=\int_0^a f(a-x)\;dx\\ y&=a-x\\dy&=-dx\\\int_0^a&\rightarrow\int_a^0\\ &\int_a^0f(y)\times-1\;dy\\&=\int_0^af(y)\;dy\\&=\int_0^af(x)dx \end{align}

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