Calculating Area of Shaded Region in Picture

[ruzfactor]

Hi

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

 

[HallsofIvy]

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 w_pk, 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- w_pk, one leg of length a, and hypotenuse of length r. By the Pythagorean theorem, we have $a^2= r^2- (r- w_{pk})^2= 2rw_{pk}- w_{pk}^2$. More importantly, the angle at the center of the circle is $arccos((r-w_{pk})/r)$ so we have a sector of a circle of radius r covering an angle of $2arccos((r- w_{pk})/r)$. The has area $2arccos((r- w_{pk}/r) r^2$.

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.