Transforming a Double Integral to a Single Integral

In summary, the conversation was about using polar coordinates to change a double integral to a single integral involving only the variable r. The given limits of integration were x^2 + y^2 = 4 in the first quadrant. The attempt at a solution resulted in the integral pi/2 sqrt(1+r^4) r dr, which was confirmed to be correct. Evaluating the integral was considered a difficult task.
  • #1
gikiian
98
0

Homework Statement



Use polar coordinates to change the following double integral to a single integral involving only the variable r.

Double-Integral( [tex]\sqrt{1+(x^{2}+y^{2})^{2}[/tex] )

The x-y region is x^2 + y^2 = 4 in the first quadrant.

2. The attempt at a solution
I got upto this:
Integral(pi/2 sqrt.(1+r^4) r dr)

Did I do it right?
 
Last edited:
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  • #2
hi gikiian! :smile:

what are the given limits of integration in the question? :confused:
 
  • #3


The x-y region is x^2 + y^2 = 4 in the first quadrant. Thanks for reminding :)
 
  • #4
gikiian said:
The x-y region is x^2 + y^2 = 4 in the first quadrant.

ah, thought so! :biggrin:

in that case, yes your integral is correct

(though showing a bit of working might have been a good idea :wink:)
 
  • #5


Hmm, thanks. Next time for sure :)

Now, evaluating the integral is a headache! :frown:
 
  • #6
(have a square-root: √ and try using the X2 tag just above the Reply box :wink:)

try the simplest possible substitution :smile:
 

1. What is the purpose of transforming a double integral to a single integral?

Transforming a double integral to a single integral allows for easier integration and calculation of the total area or volume under a curved surface. It simplifies the process by reducing the number of variables and making the integral easier to solve.

2. How do you transform a double integral to a single integral?

To transform a double integral to a single integral, you can use either the method of substitution or the method of changing the order of integration. Substitution involves replacing one variable with a function of another variable, while changing the order of integration involves swapping the order of the integration limits.

3. What are the benefits of transforming a double integral to a single integral?

Transforming a double integral to a single integral can make the integration process simpler and more straightforward. It also allows for easier visualization of the area or volume being calculated and can help in solving more complex integrals.

4. Are there any limitations to transforming a double integral to a single integral?

Transforming a double integral to a single integral may not always be possible or may lead to a more complicated integral. It is important to carefully consider the limits and variables involved before attempting to transform the integral.

5. What are some real-world applications of transforming a double integral to a single integral?

Transforming a double integral to a single integral is commonly used in physics, engineering, and economics to calculate the total volume or area of a curved surface. It is also used in statistics to calculate probabilities in multivariate distributions.

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