Area of the Curve

  • Thread starter ruzfactor
  • Start date
  • #1
82
0

Main Question or Discussion Point

Hi

How can I calculate the area of the shaded region in picture attached? Please suggest.
 

Attachments

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,772
911
The way you have shown it, the boundaries are rather complicated, and not specifically determined, functions. Do you mean to approximate them by circles? If the upper boundary, for example, is a circle with radius r, deflection wpk, and chord 2a, then we can look at the right triangle, with one vertex at the center of the circle, having one leg of length r- wpk, one leg of length a, and hypotenuse of length r. By the Pythagorean theorem, we have [itex]a^2= r^2- (r- w_{pk})^2= 2rw_{pk}- w_{pk}^2[/itex]. More importantly, the angle at the center of the circle is [itex]arccos((r-w_{pk})/r)[/itex] so we have a sector of a circle of radius r covering an angle of [itex]2arccos((r- w_{pk})/r)[/itex]. The has area [itex]2arccos((r- w_{pk}/r) r^2[/itex].

For the outer arc, you have the same thing except that instead of "r", the radius is "r+t". Use the same formula, replacing r by r+ t, and subtract the two areas to find the area you want. There is no Calculus required here.
 

Related Threads for: Area of the Curve

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
9
Views
711
Replies
15
Views
1K
  • Last Post
Replies
10
Views
4K
  • Last Post
Replies
7
Views
2K
Replies
3
Views
919
  • Last Post
Replies
1
Views
2K
Top