Find the area of the region which lies inside the graph of

and outside of the graph of

Using JKGraph:

- Click on the
*XY*button so that it changes to*Polar*. - On the
*Formulas*menu, choose*Polar Formula: R=F(@)*. - At the prompt
*R(@) =*, type in1 + cos(@) Note that the symbol @ is used to denote the angle. - Move to the prompt
*@ Starting Value*and enter -180. In this program the angle is entered in degrees. - Move to the prompt
*@ Ending Value*and enter 180. - Click on
*OK*and then you will see the graph of the cardiod. - Click on the
*2 Off*button so that we can enter two functions. The default second function is then graphed. - We want to change the default second function and so we click on
the
*1 - 2*button before we can make this change. - Again, on the
*Formulas*menu, choose*Polar Formula: R=F(@)*. - At the prompt
*R(@) =*, type in3*cos(@) - Move to the prompt
*@ Starting Value*and enter -180. - Move to the prompt
*@ Ending Value*and enter 180. - Click on
*OK*and then you will see the graph of the circle together with the cardiod which we entered above. - We calculate that the points of intersection of the two graphs; this can be done using this program (see ??). By whatever technique we use, we have that the intersections occur at the angles of -60 and 60 degrees and at R = 0.
- On the
*Domains*menu, click on*Integral Domain*. - Move to the prompt
*Integral Upper Limit:*and enter -60. - Move to the prompt
*Integral Lower Limit:*and enter 60. - Move to the prompt
*Integral Method*and choose one of the first three: lower, midpoint or upper Riemann sums. - Click on
*OK*. - Now click on the integral button and you will see the approximating
wedges. The numerical value of the approximate area appears on the
bottom bar.