Evaluate integral using Green Theorem

In summary, when integrating e^(y^2), it is important to change the order of integration and parametrize the curve along which you are integrating. This will require changing the limits of integration accordingly, making sure they are a function of y alone and not of x. Checking the result with a tool like Wolfram or Mathway can help verify the correctness of the answer.
  • #1
daphnelee-mh
66
4
Homework Statement
(question is attached below)
Relevant Equations
∮Pdx+Qdy=∬ [ (∂Q/∂x)-(∂P/∂y)]dA
1593773577298.png

I got stuck here, how to integrate e^(y^2), I searched but it's something like error function
 
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  • #2
Try changing the order of integration.
 
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  • #3
But you usually parametrize the curve along which you integrate. Edit: Change the limits of integration accordingly when you change the order of integration. These limits should be a function of y alone and not of x. Does that work?
 
Last edited:
  • #4
WWGD said:
But you usually parametrize the curve along which you integrate. Edit: Change the limits of integration accordingly when you change the order of integration. These limits should be a function of y alone and not of x. Does that work?
1593817335698.png

I got this answer after changed the integrating order
 
  • #5
daphnelee-mh said:
View attachment 265800
I got this answer after changed the integrating order
Looks right. Do you have a way of checking? Maybe Wolfram?
 
  • #6
I checked it using Mathway just now, it's correct, thank you.
 
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FAQ: Evaluate integral using Green Theorem

What is the Green Theorem?

The Green Theorem is a mathematical tool used to evaluate line integrals in two-dimensional spaces. It relates the line integral around a simple closed curve to a double integral over the region enclosed by the curve.

How is the Green Theorem used to evaluate integrals?

The Green Theorem states that the line integral of a vector field around a simple closed curve is equal to the double integral of the curl of the vector field over the region enclosed by the curve. This allows us to evaluate line integrals by converting them into double integrals, which are typically easier to solve.

What is the difference between a line integral and a double integral?

A line integral is an integral taken along a curve, while a double integral is an integral taken over a two-dimensional region. The Green Theorem allows us to convert a line integral into a double integral, making it easier to evaluate.

What are the requirements for using the Green Theorem?

The Green Theorem can only be used for line integrals in two-dimensional spaces. Additionally, the curve must be simple and closed, meaning it does not intersect itself and encloses a finite area.

Can the Green Theorem be used for any type of curve?

No, the Green Theorem can only be used for simple and closed curves in two-dimensional spaces. It cannot be used for curves that intersect themselves or curves in higher dimensions.

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