Area of a polar function

In summary, the student is trying to find the area enclosed by a polar curve using the polar area formula. They are using Maple to plot the function and are struggling with simplifying the integral. Some tips are provided to help simplify the equation and find the enclosed area, such as expanding trigonometric functions, finding symmetries, using trigonometric identities, and using numerical methods if needed.
  • #1
tomcochrane
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Homework Statement


I'm trying to find the area enclosed by the polar curve [tex]r^2 = 2-cos(r*cos(\theta))*sin(r*sin(\theta))[/tex].


Homework Equations



I've been trying to use the polar area formula, [tex] A= \frac{1}{2}\int _a ^b r^2 d\theta[/tex], where r is a function of [tex]\theta[/tex]



The Attempt at a Solution



I'm using Maple, and it's getting a little messy. First I had to plot the function, and that was fine. For the plot I used r goes from -2 to 2 and [tex]\theta[/tex] goes from 0 to [tex]\pi[/tex]. That was fine.

Now I have to find the area of the plot. I tried getting the r's and theta's onto separate sides of the equation, so I could put r into the area integral, but you could imagine that would be a huge mess. I must be forgetting something. Where should I go from there? Or am I missing something and heading in the wrong direction? Thanks.
 
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  • #2




Hi there, great question! It looks like you're on the right track with using the polar area formula. Here are a few tips to help you simplify the integral and find the enclosed area:

1. First, try to simplify the equation by expanding the trigonometric functions. This will help you get rid of any nested functions and make the equation easier to work with.

2. Next, try to find any symmetries in the equation. For example, does the function have any symmetry about the origin or any other point? This can help you simplify the integral by reducing the limits of integration.

3. You can also try using trigonometric identities to simplify the integral. For example, the identity cos^2(x) + sin^2(x) = 1 can be useful in this case.

4. If all else fails, you can always use a numerical method like Simpson's rule to approximate the area under the curve.

I hope this helps and good luck with your calculations!
 

What is the definition of "area of a polar function"?

The area of a polar function refers to the total space enclosed by a polar curve on a polar coordinate system. It is also known as the polar area or the area under a polar curve.

How do you calculate the area of a polar function?

The area of a polar function can be calculated using the formula A = ½∫ab(r(θ))2dθ, where r(θ) is the polar function and a and b are the limits of integration.

What is the difference between finding the area of a polar function and a regular function?

The main difference is that a polar function uses polar coordinates (r,θ) instead of Cartesian coordinates (x,y). This means that the area under a polar curve is calculated using a different formula and involves integration with respect to θ instead of x.

Can the area of a polar function be negative?

No, the area of a polar function cannot be negative. It represents the total positive space enclosed by the polar curve. If the integral produces a negative value, it means that the area is being counted in the opposite direction and should be taken as the absolute value.

What are some real-world applications of finding the area of a polar function?

Finding the area of a polar function is useful in many fields such as engineering, physics, and astronomy. It can be used to calculate the area of a cross-section in 3D printing, the moment of inertia of rotating objects, and the mass distribution of planets and stars.

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