Problem:
Estimate the area of the region which lies inside the loop of the graph of the curve given by the parametric equations:

x = f(t) = t2 + 1
y = g(t) = t3 - t + 3

Visualization:
Using JKGraph:

  1. Click on the XY button so that it changes to Param.
  2. On the Formulas menu, choose Parametric Formulas: X=F(T) & Y=G(T).
  3. At the prompt X(T) = , type in
    t^2+1
    and at the prompt Y(T) = , type in
    t^3-t+3
  4. Move to the prompt T Starting Value and enter -2.
  5. Move to the prompt T Ending Value and enter 2.
  6. Click on the World Graph Domain button.
  7. Move to the prompt Graph X Minimum Value: and enter 0.
  8. Move to the prompt Graph Y Minimum Value: and enter -1.
  9. Move to the prompt Graph X Maximum Value: and enter 5.
  10. Move to the prompt Graph Y Maximum Value: and enter 5.
  11. Click on the OK button.
  12. Click on the Edit Integral Domain button.
  13. If necessary, move to the prompt Integral Upper Limit and enter 1.
  14. If necessary, move to the prompt Integral Lower Limit and enter 0.
  15. Set the Number of Subinterals to be 10.
  16. Choose the Integral Method.
  17. Click on the OK button twice and then you will see the graph of the curve.
  18. Click on the Integral button and you will see the following.

    Note that the approximate area, given at the bottom of the screen, is 1.3701.

  19. On the Domains, pick Integral Domain.
  20. Move to the prompt Integral Upper Limit and enter 0.
  21. Move to the prompt Integral Lower Limit and enter -1.
  22. Click on the OK button and then you will see the graph of the curve.
  23. Click on the Integral button and you will see the following.

    Note that the approximate area, given at the bottom of the screen, is -1.6616; it follows that the desired approximation of the area inside the loop is 1.6616 - 1.3701 = .2915.