Evaluate the Following Integral

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Homework Help Overview

The discussion revolves around evaluating a double integral involving the function sin(2y^2) with respect to the variables x and y. Participants are attempting to understand the limits of integration and the proper approach to solving the integral.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need to switch the limits of integration and question the interpretation of the bounds, particularly the relationship between y and x. There are attempts to clarify whether y is dependent on x or treated as a dummy variable.

Discussion Status

The discussion is active with various interpretations being explored. Some participants have suggested graphical methods to understand the limits, while others are focused on the integration process itself. There is no explicit consensus on the approach, but several lines of reasoning are being examined.

Contextual Notes

There are mentions of difficulties with integrating sin(2y^2) and the need for clarity on the limits of integration. Some participants express confusion regarding the attachment of the integral question, which may affect their understanding.

Northbysouth
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Homework Statement


Evaluate the following integral.

I have attached an image of the question.


Homework Equations





The Attempt at a Solution


I know that I need to switch the limits, but I'm not sure how to do this.

Do I just solve y = x/2 for x which would give me:

x = 2y and substitute this in place of the x/2?

Help would be appreciated.
 

Attachments

  • math 2224 14.2 1.png
    math 2224 14.2 1.png
    5.3 KB · Views: 602
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Where does it say ##y = \frac{x}{2}##? I believe in your case, y is independent of x.
 
I assumed that that was what the lower bound of the integral with respect to y is.
 
Northbysouth said:
I assumed that that was what the lower bound of the integral with respect to y is.

That is part of the limit of integration. It doesn't tell us that y is a function of x. y in this case is just like x; it's a dummy variable. Just integrate first with respect to y, then evaluate the inner integral. At this point, the integrand should be in terms of x. Then just integrate again and evaluate.
 
The problem is, I can't figure out how to integrate sin(2y^2) because u substitution doesn't work. When I spoke with my professor he said to switch the limits but I am not sure how to do this.
 
I've got it now. Using the boundaries for y and x I draw a graph of the boundaries which gave me a triangle bounded by the sides X=0, Y=1 and y=x/2

Then I rearranged the integral so that I integrated with respect to x before I integrated integrated with respect to y.

In the end I got the answer to be

-1/2 + cos(2)/2
 
Northbysouth said:

Homework Statement


Evaluate the following integral.

I have attached an image of the question.


Homework Equations





The Attempt at a Solution


I know that I need to switch the limits, but I'm not sure how to do this.

Do I just solve y = x/2 for x which would give me:

x = 2y and substitute this in place of the x/2?

Help would be appreciated.

I could not read your attachment on my i-Phone 4, so here it is again in readable form:
[tex]\int_0^2 \int_{x/2}^1 sin(2y^2)\, dy \, dx.[/tex]

RGV
 
Sorry, my misinterpretation/ignorance.
 

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