Reverse the order of integration?

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The discussion focuses on reversing the order of integration for a double integral originally set as dy dx with specified limits. The user attempted to change it to dx dy but received feedback indicating that the new limits were incorrect. A graphical representation of the region is suggested to clarify the bounds, emphasizing that the relationship between x and y must be properly understood. It is highlighted that the equation y = e^x does not imply x = e^y, and the correct transformation requires solving for x in terms of y. Understanding the geometric interpretation of the integration limits is crucial for accurately reversing the order of integration.
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Homework Statement



Reverse the order of integration?
Original has it in the form of dy dx, 0<=x<=ln6, 1<=y<=e^x.
I made it dx dy, with 1<=x<=e^y, 0<=y<=ln6
Is this right?

Homework Equations





The Attempt at a Solution

 
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Draw a graph. x ranges between 0 and ln(6) so draw two vertical lines there. For each x[/itex], y ranges from 1 to y= e^x do draw the horizontal line y= 1 and the y= e^x. Notice that e^0= 1 so that forms something that looks like a kind of "right triangle" with a curved hypotenuse.

Now, what are the lowest and highest value of y in that "triangle"? Saying that x< ln 6 does NOT mean y< ln 6. y is not x!
Imagine a horizontal line at some y-value. x ranges from the left end of that line to the right end. What are those values?

y= e^x does NOT give x= e^y! Solve y= e^x for x.
 
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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