Example 1

The graph of the polar equation is a cardiod:

> P1 := polarplot( 1+cos(theta), theta=0..2*Pi ): P1;

>

The graph of is recognized as a line. To see exactly what line we can convert back to Cartesian coordinates

> eq1 := tan(theta)=y/x;

> eq2 := eval( eq1, theta=Pi/6 );

> eq3 := isolate( eq2, y );

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or we can plot the curve as a parametric polar curve (parameterized by r):

> P2 := polarplot( [r,Pi/6,r=0..5] ): P2;

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> display( [P1,P2] );

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The smaller region is seen to be the one described by:

0 <= <=

<= <=

Thus, the area of the smaller region can be found by evaluating

> A := Int( Int( r, r=0..1+cos(theta) ), theta=Pi/6..Pi );

> value( A );

>