# Area of the Curve

Hi

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

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• plate.jpg
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## Answers and Replies

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 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 $a^2= r^2- (r- w_{pk})^2= 2rw_{pk}- w_{pk}^2$. 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 $2arccos((r- w_{pk}/r) r^2$.