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 LL is the width of the rectangle, and the curved length of a semicircle is πr \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 πr \pi r . For each horizontal strip, we have an area element (technically length element) of LL . Hence, the area is

RLdR=πr×L. \int_{R} L \, dR = \pi r \times L.

What is the area of the shaded figure?


Inspiration, see solution comments.

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