Double integral to find the area of the region enclosed by the curve

In summary, to find the area of the region enclosed by the curve r=4+3cos(theta), you need to draw a polar curve and determine the minimum value of r to be 1. Then, the range for r is 1 to 4+3cos(theta), and the range for theta is 0 to pi. You can multiply the answer by 2 to obtain the full area. It is also helpful to plot the curve to understand why the limits for theta are 0 and pi, rather than 0 and 2pi. The curve is a cardioide.
  • #1
PhysicsMajor
15
0
Greetings all,

I need help setting up this problem:

Use a double integral to find the area of the region enclosed by the curve

r=4+3 cos (theta)

Thanks
 
Physics news on Phys.org
  • #2
Hello.

1)You'll have to draw a polar curve to help you out with this question. From the drawn polar curve, you'll get the minimum value of r to be 1. (when cos(theta) is negative)
2) Thus it follows that the range for r is 1<=r<=4+3cos(theta)
Hence we'll integrate r from 1 to 4+3cos(theta)
3) For the range of theta, you can use the range from 0 to pi for simplicity in calculations.(Just multiply the answer by 2 to obtain the full area.)

I hope this helps =)
 
  • #3
Yeah,a plot might help u convince why the limit wrt [itex] \theta [/itex] need to be 0 and [itex] \pi [/itex] and why you shouldn't integrate from 0 to [itex] 2\pi [/itex]

I think it's a cardioide.


Daniel.
 
Last edited:

Related to Double integral to find the area of the region enclosed by the curve

1. What is a double integral?

A double integral is a mathematical concept used to find the area of a region bounded by a curve in two dimensions. It involves calculating the sum of infinitely small rectangular areas within the given region.

2. How is a double integral different from a single integral?

A single integral is used to find the area under a curve in one dimension, while a double integral is used to find the area within a curve in two dimensions.

3. What is the process for using a double integral to find the area of a region?

The process involves setting up the integral with the appropriate limits of integration based on the given curve and then solving the integral using various integration techniques.

4. Can a double integral be used to find the area of any region?

Yes, as long as the region can be represented by a mathematical function or a set of equations, a double integral can be used to find its area.

5. Are there any limitations to using a double integral to find the area of a region?

While a double integral can accurately find the area of most regions, there are certain complex shapes or regions with irregular boundaries that may require more advanced mathematical techniques to find the area accurately.

Similar threads

Replies
2
Views
2K
Replies
20
Views
2K
Replies
2
Views
420
Replies
2
Views
436
  • Calculus
Replies
24
Views
3K
  • Calculus
Replies
29
Views
934
  • Calculus
Replies
5
Views
2K
Replies
3
Views
1K
Replies
4
Views
2K
Replies
3
Views
484
Back
Top