   Chapter 7.P, Problem 5P

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# An ellipse is cut out of a circle with radius a. The major axis of the ellipse coincides with a diameter of the circle and the minor axis has length 2b. Prove that the area of the remaining part of the circle is the same as the area of an ellipse with semiaxes a and a − b .

To determine

To prove:

That the area of the remaining part of the circle is the same as the area of an ellipse with semi axes a and ab.

Explanation

Given:

The area of the ellipse with major axis 2a and the minor axis 2b is xdy

Formulae used:

The integration method

Any point on this ellipse can be written as (acosθ,bsinθ)

Therefore, the area of the ellipse is

A=02πabcos2θdθ=ab02πcos2θ</

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