Calculating Area with Double Integrals - Solving for Unknown Functions

In summary, the author was studying for a calculus final and stumbled upon a problem that involved finding the area bounded by a function. However, the graph was unknown to the author, so substitution and polar coordinates were attempted but did not work. A change of variables to u and v solved the problem.
  • #1
Lorenc
8
0
"Hey guys, how are you? I was studying for my calculus final and stumbled upon a peculiar function.

Homework Statement



Now I have to find the area bounded by the function (x^2+y^2)^3=xy^4 using a double integral. Now, the problem is that the graph is totally unknown to me (I have some ideas but I am not shure).

Homework Equations



(x^2+y^2)^3=xy^4

The Attempt at a Solution



A substitution with u and v, doesn't seem to work and going to polar doesn't work either :/ Maybe I am doing something wrong, I don't know. Can anybody help me? Thank you in advance :)
 
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  • #2
Use polar coordinates.
 
  • #3
It doesn't seem to solve that way. Can you please write just the polar equation in this case?
 
  • #4
Lorenc said:
"Hey guys, how are you? I was studying for my calculus final and stumbled upon a peculiar function.

Homework Statement



Now I have to find the area bounded by the function (x^2+y^2)^3=xy^4 using a double integral. Now, the problem is that the graph is totally unknown to me (I have some ideas but I am not shure).

Homework Equations



(x^2+y^2)^3=xy^4

The Attempt at a Solution



A substitution with u and v, doesn't seem to work and going to polar doesn't work either :/ Maybe I am doing something wrong, I don't know. Can anybody help me? Thank you in advance :)

Just as a matter of terminology: you do not have a "function; you have two functions and one equation connecting them (to form a curve). At first I had a lot of trouble trying to decipher your post.

Certainly, a judicious change of variables makes the problem pretty straightforward.
 
  • #5
Two functions? Yes, but can the whole equation be plotted using the sepparate functions? I am sorry, but I really need to imagine the area of integration. And as for the change of variables, I was thinking u = x^2 + y^2, ok, but then?
 
  • #6
I am attempting to do this problem, quick question just for clarity: is it x*y^4 of (x*y)^4?
 
  • #7
Jufro said:
I am attempting to do this problem, quick question just for clarity: is it x*y^4 of (x*y)^4?

It is read ##x*y^4##.

As for the problem, a simple change to polar co-ordinates is all that is needed.
 
  • #8
Thank you everyone :)
 

Related to Calculating Area with Double Integrals - Solving for Unknown Functions

What is a double integral?

A double integral is a type of mathematical operation used to find the area between a two-dimensional region and the x-y plane. It involves integrating a function of two variables over a given region in the x-y plane.

How do you set up a double integral to find area?

To set up a double integral, you first need to determine the limits of integration for both the x and y variables. These limits will define the region over which the function will be integrated. Then, you can write the integral in the form of ∫∫f(x,y)dA, where f(x,y) is the function being integrated and dA represents the infinitesimal area element within the region.

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

A single integral involves integrating a function of one variable over a given interval, while a double integral involves integrating a function of two variables over a given region. The result of a single integral is a number, while the result of a double integral is a number representing a 2-dimensional area.

What is the significance of the order of integration in a double integral?

The order of integration in a double integral refers to the order in which the variables are integrated. This can affect the difficulty of evaluating the integral and can also change the limits of integration. It is important to choose the correct order of integration in order to accurately calculate the area.

Can a double integral be used to find the volume of a 3-dimensional object?

Yes, a double integral can be used to find the volume of a 3-dimensional object in certain cases. This is done by integrating a function of two variables over a given region in the x-y plane, where the function represents the height of the object at each point. However, this method is limited to certain types of objects and may require additional mathematical techniques.

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