Breaking The Disc Method

Calculus Level 3

The above figure consists of a rectangle with a semicircle cut out of one end and added to the other end, where \(L\) is the width of the rectangle, and the curved length of a semicircle is \( \pi r \).

To calculate the area of the shaded figure, Svatejas applies the disc method as follows:

Consider the axis of integration to be the semicircular arc, which has length \( \pi r \). For each horizontal strip, we have an area element (technically length element) of \(L \). Hence, the area is

\[ \int_{R} L \, dR = \pi r \times L. \]

What is the area of the shaded figure?

Inspiration, see solution comments.


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