Polar Regions: Area, Arc Length, and Surface Area

In summary: So I'm a bit confused.In summary, to find the area inside the large loop minus the area of the small loop, we need to use the formula A = \int\frac{1}{2}(f(\Theta))^{2}d\Theta with upper limit 2π/3 and lower limit 4π/3. For the arc length of the small loop, we use the formula AL = \int\sqrt{r^{2}+(\frac{dr}{d\Theta})^{2}} d\Theta with upper limit 4π/3 and lower limit 2π/3. The formula for finding the surface area of the surface formed by revolving the large loop about the initial ray is not provided,
  • #1
spacetime24
4
0

Homework Statement



Consider the graph (see attachment) of r = 1 +2cos[tex]\Theta[/tex] in polar coordinates. SET UP integrals to find
1. the area inside the large loop minus the area of the small loop.
2. the arc length of the small loop
3. the surface area of the surface formed by revolving the large loop about the initial ray.


Homework Equations



area A of the polar region
A = [tex]\int[/tex][tex]\frac{1}{2}[/tex](f([tex]\Theta[/tex]))[tex]^{2}[/tex]d[tex]\Theta[/tex] with upper limit b and lower limit a.

arc length AL of the polar region:
AL = [tex]\int[/tex][tex]\sqrt{r^{2}+(\frac{dr}{d\Theta})^{2}} d\Theta[/tex] with upper limit b and lower limit a.

not sure what equation i need to figure out the surface area one

The Attempt at a Solution



I'm pretty much lost when it comes to the entire problem, and have no ideas where to start. Please help!
 

Attachments

  • Calc Graph.jpg
    Calc Graph.jpg
    16.8 KB · Views: 450
Physics news on Phys.org
  • #2
You've got the equations right. You just have to figure out what to use for the bounds. For 1, if you want to encompass only the larger arc, what should the upper bound be?
 
  • #3
Well if the 4π/3 ray moved in a counterclockwise motion until the 2π/3 ray, then the ray would sweep the entire area of the large loop. But that would mean doing an integral from a lower bound of 4π/3 to the upper bound of 2π/3. This to me doesn't seem right since usually the lower bound is smaller than the upper bound.
 

1. What is the difference between area, arc length, and surface area in polar regions?

In polar regions, area refers to the size of a two-dimensional space within a specific boundary. Arc length refers to the distance between two points along a curved surface, such as the circumference of a circle. Surface area, on the other hand, refers to the total area of all the faces of a three-dimensional object, including the curved surface.

2. How is the area of a polar region calculated?

The area of a polar region can be calculated using the formula A = 1/2 * r^2 * θ, where r is the radius of the region and θ is the angle formed by two radii at the center of the region. This formula is based on the formula for the area of a sector of a circle.

3. What is the significance of arc length in polar regions?

Arc length is important in polar regions because it helps measure the distance between two points along a curved surface, such as the distance traveled along a meridian or longitude line. This can be useful for navigation and mapping purposes.

4. How does surface area impact polar regions?

The surface area of polar regions is an important factor in understanding the climate and environment of these regions. It can affect factors such as heat absorption and distribution, which can have a significant impact on the weather and wildlife in these regions.

5. Can the formulas for calculating area, arc length, and surface area be applied to other regions besides polar regions?

Yes, the formulas for area, arc length, and surface area can be applied to any region or shape, as long as the appropriate measurements and angles are known. These formulas are widely used in mathematics and science to calculate the size and dimensions of various objects and spaces.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
848
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
487
Replies
14
Views
939
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
589
  • Calculus and Beyond Homework Help
Replies
7
Views
3K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
Back
Top