Iterated Integral: Changing the Order of Integration

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In summary, the conversation is about evaluating an iterated integral and changing the order of integration in order to evaluate the resulting integral. The first part involves sketching the region over which integration takes place, and the second part involves finding the correct integral by differentiating and checking the work.
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s10dude04
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Homework Statement



(a) For the iterated integral \\cos(x/y)dydx (inner limits x to 1, outer limits 0 to 1) sketch the region in the plane corresponding to the double integral this interated integral evaluates.

(b) Evaluate the double integral by changing the order of integration in the iterated integral and evaluating the resulting iterated integral.

Homework Equations




The Attempt at a Solution



Taking the integral of cos(x/y) with respect to y gives you sin(2x/y^2) no?

Then you evaluate from x to 1, which would be sin(2x)-sin(2/x)?

Then doing the integral of that gives ********?

Am I heading in the right direction?
 
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  • #2
s10dude04 said:

Homework Statement



(a) For the iterated integral \\cos(x/y)dydx (inner limits x to 1, outer limits 0 to 1) sketch the region in the plane corresponding to the double integral this interated integral evaluates.

(b) Evaluate the double integral by changing the order of integration in the iterated integral and evaluating the resulting iterated integral.

Homework Equations




The Attempt at a Solution



Taking the integral of cos(x/y) with respect to y gives you sin(2x/y^2) no?

Then you evaluate from x to 1, which would be sin(2x)-sin(2/x)?

Then doing the integral of that gives ********?

Am I heading in the right direction?
Not at all, as far as I can see from your work. The first part asks you to sketch the region over which integration takes place. Have you done that? This region can be described pretty simply.
For the second part, how to you go from [itex]\int cos(x/y)dy[/itex] to sin(2x/y^2)? If you check this work by differentiating sin(2x/y^2) with respect to y, do you get cos(x/y)?
 

1. What is an iterated integral?

An iterated integral is a type of multiple integral that involves integrating a function over a specific region in multiple stages. It is typically used to calculate the volume, area, or other quantities of a three-dimensional region.

2. What is the difference between a single integral and an iterated integral?

A single integral involves integrating a function over a one-dimensional interval, while an iterated integral involves integrating a function over a two-dimensional or three-dimensional region. In other words, an iterated integral is a generalization of a single integral.

3. How do you set up an iterated integral?

To set up an iterated integral, you first need to determine the limits of integration for each variable. This involves understanding the bounds of the region over which the function is being integrated. Then, you write the integral with the innermost integral representing the integration with respect to one variable, and the outer integrals representing the integration with respect to the other variables.

4. What is the purpose of using an iterated integral?

The purpose of using an iterated integral is to calculate quantities such as volume, area, or mass for three-dimensional regions. It can also be used to calculate other types of integrals, such as double and triple integrals, which have important applications in physics, engineering, and economics.

5. What are some common applications of iterated integrals?

Iterated integrals have many applications in various fields, including physics, engineering, economics, and statistics. They are commonly used to calculate volumes of three-dimensional objects, surface areas, moments of inertia, centers of mass, and probabilities in multivariate distributions.

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