How to find the limit if integration of polar curves?

In summary: The graph will show you how the curve looks and how it behaves as ##\theta## varies from 0 to 2##\pi##. This will give you a much better understanding of the problem and help you to answer the questions that have been asked.
  • #1
realism877
80
0

Homework Statement



r=3+2cosθ


Homework Equations







The Attempt at a Solution



The text shows that it's from 0 to 2pi.

How did it come to those limits without graphing?

I set r=o. What do I do from there?
 
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  • #2
What's the question you want to answer by which integration? If you don't ask a question (also to yourself!) you'll have big trouble to find an answer, because there is no question ;-).
 
  • #3
realism877 said:

Homework Statement



r=3+2cosθ


Homework Equations







The Attempt at a Solution



The text shows that it's from 0 to 2pi.

How did it come to those limits without graphing?

You DO graph it.
 
  • #4
LCKurtz said:
You DO graph it.

How do you find it on the calculator?
 
  • #5
realism877 said:
How do you find it on the calculator?

How do you find what on a calculator? Graph it with your calculator or by hand.
 
  • #6
realism877 said:
How do you find it on the calculator?

As vanhees71 says, you need to state the complete problem. You have not provided enough information to find the limits. Are you asked for the area enclosed?
 
  • #7
haruspex said:
As vanhees71 says, you need to state the complete problem. You have not provided enough information to find the limits. Are you asked for the area enclosed?

Find the area of the curve r=3+2cosθ

Find the area it encloses.
 
  • #8
You could first think about a simpler problem, namely finding the area of a rectangle by integration. Suppose one corner of the rectangle is at the origin (0,0) and the opposite corner is at point (a,b). Now the area is

[itex]A=\int^{b}_{0}\int^{a}_{0}dxdy[/itex]

In your problem, we are integrating over a region of (r,##\theta##)-plane. Now you should consider the following questions:

1. What's the equivalent of the area element ##dxdy## in polar coordinates?

2. What are the integration limits in your problem (the upper integration limit when integrating with respect to r coordinate is going to be a function of ##\theta##)
 
  • #9
realism877 said:
Find the area of the curve r=3+2cosθ

Find the area it encloses.
If it encloses an area then there must be two values of theta that produce the same r. Since you only want one copy of the area, you need to find two consecutive such thetas. In general, you might then have to worry whether different consecutive pairs produce different areas, but you should be able to show easily that does not happen here.
 
  • #10
Regardless of the other suggestions you are receiving, your first step should be to GRAPH IT.
 

1. How do I determine the limits of integration for a polar curve?

The limits of integration for a polar curve are determined by the range of values for the variable theta. These values can be found by looking at the graph of the polar curve and identifying the starting and ending points for theta. Alternatively, you can also solve for the intersections of the curve with the horizontal and vertical axes.

2. What is the process for finding the area enclosed by a polar curve?

To find the area enclosed by a polar curve, you first need to determine the limits of integration as described above. Then, use the formula A = ½∫(r^2)dθ, where r is the polar function and dθ represents the differential element of theta. Integrate the function between the limits of integration to find the area.

3. How do I know if a polar curve is symmetric about the origin?

A polar curve is symmetric about the origin if it satisfies the condition r(θ) = r(-θ) for all values of theta. This means that the distance from the origin to a point on the curve is the same as the distance from the origin to the point reflected across the x-axis. You can also determine symmetry by looking at the graph of the polar curve.

4. Can I use the fundamental theorem of calculus to find the limit of integration for polar curves?

Yes, the fundamental theorem of calculus can be used to find the limit of integration for polar curves. This is because the theorem states that the integral of a function can be evaluated by finding its antiderivative and evaluating it between the limits of integration. However, you may need to use the substitution method or other techniques to find the antiderivative of the polar function.

5. How do I convert a polar curve into rectangular coordinates?

To convert a polar curve into rectangular coordinates, you can use the following formulas: x = rcos(θ) and y = rsin(θ). Simply plug in values for the variable theta to get corresponding values for x and y. Keep in mind that the limits of integration may also need to be adjusted when converting to rectangular coordinates.

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