Average value of f(x,y) = xy in quarter circle x^2 + y^2 < 1 in Q1.

In summary, the problem is to find the average value of the function f(x,y) = xy for the quarter circle x^2 + y^2 ≤ 1 in the first quadrant. Using the average value equation and the area of a quarter of a unit circle, the average value is calculated to be 1/2π. Posting individual problems and stating them clearly will likely yield faster and more helpful responses.
  • #1
VinnyCee
489
0
Here is the problem:

Find the average value of [tex]f\left(x, y\right) = x\;y[/tex] for the quarter circle [tex]x^2 + y^2 \le 1[/tex] in the first quadrant.

Here is what I have:

Average value equation is [tex]\frac{1}{Area\;of\;R} \iint_{R} f\left(x, y\right) dA[/tex]

[tex]f\left(x, y\right) = x\;y = \left(r\;\cos\theta\right)\left(r\;\sin\theta\right)[/tex]

The area of one quarter of a unti circle is [tex]\frac{\pi}{4}[/tex], right?

[tex]Average = \frac{4}{\pi}\;\int_{0}^{\frac{\pi}{2}}\int_{0}^{1}\;\left(r\;\cos\theta\right)\left(r\;\sin\theta\right)\;r\;dr\;d\theta = \frac{1}{2\pi}[/tex]

Is this correct?
 
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  • #2
Absolutely, and good job remembering the jacobian in polar coordinates.
 
  • #3
Thank you

Many thanks. I will probably be checking a few others before I am through.

I wonder if I should make a thread for "Check my Calculus Answers Please" and put them all in there or something? On the other hand, I think that individually posting each problem gets quicker results and is easier to search (i guess).
 
  • #4
In order to get useful feedback and responses, it's best to title your threads appropriately--- putting something like "check my work..." generally isn't too appealing, so you're probably right--- that is, make threads that are individual for a specific problem, with the statement and reasoning placed in the first post.

You have been doing this but just so others know!
 

Related to Average value of f(x,y) = xy in quarter circle x^2 + y^2 < 1 in Q1.

1. What is the average value of f(x,y) in quarter circle x^2 + y^2 < 1 in Q1?

The average value of f(x,y) in quarter circle x^2 + y^2 < 1 in Q1 is the sum of all the values of f(x,y) within the quarter circle, divided by the total number of points. This can be calculated by finding the integral of f(x,y) over the quarter circle and dividing by the area of the quarter circle.

2. How do you find the integral of f(x,y) over the quarter circle?

To find the integral of f(x,y) over the quarter circle, you can use a double integral with the limits of integration being -1 to 0 for x and 0 to 1 for y. This will integrate the function over the quarter circle and give you the total sum of the values within the quarter circle.

3. Can you explain how the average value of f(x,y) is affected by the shape of the quarter circle?

The shape of the quarter circle will affect the average value of f(x,y) because it determines the area over which the function is being integrated. A larger quarter circle will have a larger area and therefore a larger average value of f(x,y) compared to a smaller quarter circle with a smaller area.

4. How does the average value of f(x,y) in quarter circle x^2 + y^2 < 1 in Q1 compare to the average value in the entire circle?

The average value of f(x,y) in quarter circle x^2 + y^2 < 1 in Q1 will be equal to one fourth of the average value in the entire circle. This is because the quarter circle represents one fourth of the total area of the circle, so the average value will also be one fourth of the total average value.

5. How can the average value of f(x,y) be used in real-world applications?

The average value of f(x,y) can be used in real-world applications to calculate the average value of a physical quantity over a specific area. For example, it can be used to find the average temperature in a quarter circle region on a map, or the average rainfall over a specific quadrant of a city. It can also be used in physics and engineering to find the average force or energy over a certain region.

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