Changing the limits of integration

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The discussion focuses on understanding the transformation of limits in integrals. The user is confused about how two integrals relate to a third integral after changing the limits. It is clarified that swapping the limits of integration changes the sign of the integral. The substitution method, specifically u = -x, is suggested as a way to change the limits from -π to π. This approach resolves the user's confusion regarding the integration limits.
jinksys
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! SOLVED !

This isn't a homework problem, just an equation in my chapter. I don't see how the two integrals pointed to by the blue arrows become the integral pointed to by the red arrow. I know that if you swap the limits of integration, you change the sign of the integral. However, how do you change the -pi to pi in the second integral?

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Use the substitution u=-x
 
Of course, thanks.
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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