6.1 Areas between Curves

1-22. Find the area of the region.


1. .

 .


2. .

 .


3. .

 .


4. .

 .


5. .

 .


6. .


 .


8. .

 .


9. .

 .


10. .

 .


11. .

 .


12. .

 .


13. .

 .


14. .

 .


15. .

 .


16. .

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17. .

 .


18. .

 .


19. .

 .


20. .

 .


21. .

 .


22. .

 .


23-24. Find the integral and interpret it as the area of a region.


23. .

 .


24. .

 .


25-26. Approximate the area of the region bounded by the given curves using the Midpoint Rule with .


25. .

 .


26. .

 .


27-30. Determine the area of the region bounded by the curves.


27. .

 .


28. .

 .


29. .

 .


30. .

 .


31. Find the region defined by the inequalities .

 .


32. Find the area enclosed by the loop of the curve with equation .

    (Tschirnhausen's cubic.)

 .


33. Find the area of the region bounded by the curve , the tangent line to this curve at , and the -axis.

 .


34. Find the number such that the line divides the region bounded by the curves and into two regions with equal area.

   .


35. (a) Find the number such that the line bisects the area under the curve

        .

    (b) Find the number such that the line bisects the area in part (a).

   .

        .


36. Find the values of such that the area of the region enclosed by the parabolas and is 1944.

   .

      For and is another solution.

      Therefore .


37-41. Find the area of the region bounded by the given curves.


37. , .

Since ,


38. , .

Since


39. , .

Since

40. , .



41. , , .