# Reverse the order of integration?

• turbokaz
That is the formula for the upper boundary of the region if you integrate with respect to x first.In summary, the conversation discusses reversing the order of integration for a given region where x ranges from 0 to ln6 and y ranges from 1 to e^x. One person suggests changing the order to dx dy with x ranging from 1 to e^y and y ranging from 0 to ln6. The other person advises drawing a graph to visualize the region and determining the correct boundaries for x and y. They also clarify that y is not equal to x and remind to solve for x when given y= e^x.
turbokaz

## 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?

## The Attempt at a Solution

Wow, you waited 12 whole minutes before bumping? That's a good way to get banned from this forum!

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.

## 1. What is "reverse the order of integration"?

"Reverse the order of integration" is a mathematical technique used in multiple integrals to change the order in which the integrals are evaluated. This can sometimes make the integration process easier or more efficient.

## 2. When should "reverse the order of integration" be used?

"Reverse the order of integration" should be used when the original order of integration is difficult or impossible to evaluate. By changing the order, the integral may become simpler to solve or may lead to a more efficient solution method.

## 3. How do you perform "reverse the order of integration"?

To perform "reverse the order of integration," you need to first identify the limits of integration for each variable. Then, simply switch the order in which the variables are integrated, while keeping the limits of integration for each variable the same. This can be done by changing the order of the nested integrals and adjusting the limits of integration accordingly.

## 4. Are there any limitations to "reverse the order of integration"?

Yes, there are some limitations to "reverse the order of integration." It can only be used for certain types of integrals, such as double or triple integrals, and the integrand must be continuous over the region of integration. Additionally, the order of integration cannot always be reversed without changing the limits of integration, which may make the integral more difficult to evaluate.

## 5. Can "reverse the order of integration" be used for any type of function?

No, "reverse the order of integration" cannot be used for any type of function. It is most commonly used for functions that are continuous and well-behaved, such as polynomial functions, trigonometric functions, and exponential functions. It may not be possible to use this technique for more complex or discontinuous functions.

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