Breaking The Disc Method

Calculus Level 2

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?

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