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Problems. Integral calculus
1. Evaluate the indefinite integral:. 2. Evaluate the indefinite integral:. 3. Evaluate the indefinite integral:. 4. Evaluate the indefinite integral:.
.5. Evaluate the indefinite integral:.
6. Evaluate the indefinite integral:. 7. Evaluate the indefinite integral:. 8. Evaluate the indefinite integral:. 9. Evaluate the indefinite integral:. 10. Evaluate the indefinite integral:. 11. Evaluate the indefinite integral:. 12. Evaluate the indefinite integral:. 13. Evaluate the indefinite integral:.
14. Evaluate the indefinite integral:. 15. Evaluate the indefinite integral:. 16. Evaluate the indefinite integral:. 17. Evaluate the indefinite integral by per partes method:. 18. Evaluate the indefinite integral by per partes method:. 19. Evaluate the indefinite integral:. 20. Evaluate the indefinite integral:. 21. Evaluate the indefinite integral:. 22. Evaluate the definite integral:. 23. Evaluate the definite integral:. 24. Evaluate the definite integral:. 25. Evaluate the definite integral:. 26. Evaluate the definite integral:. 27. Evaluate the definite integral:.
Solutions
1.
Absolute value can be omitted since for all.
We obtain an equation:
from which:
28. Surface area of the region bounded by the curves: and
can be obtained as definite integral:.
First we find the crossing points of the curves:
Fig. 1. Region bounded by the curves: and.
29. Surface area of the region bounded by the symmetric curve:
can be calculated as 4 times the region located in the first quadrant:
Fig. 2. Region bounded by the curve:
30. Volume generated by revolving the area bounded by the parabola: about the
-axis for can be calculated as:
33. The length of the arc of the curve: on the interval will
be calculated as:
Fig. 5. Graph of a curve:
34. The area of the surface of revolution generated by revolving about -axis the arc of the
ellipse: with the length of its main axis equal to 4 will be calculated as:
.
First we evaluate:
