Using the law of reciprocity show that among all the closed curves that border a given area, the shortest is the circle. This is a calculus of variation problem. Please show me complete solution with details.

Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Polynomial And Rational Functions
Section9.4: Graphing Polynomial Functions
Problem 43PS
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Using the law of reciprocity show that among all the closed curves that border a given area, the shortest is the circle.

This is a calculus of variation problem.

Please show me complete solution with details.

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