Questions and Answers

PART - A 

QUESTIONS AND ANSWERS

1. Evaluate

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2. Evaluate

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3. Evaluate

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4. Evaluate 

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5. Evaluate 

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6. Evaluate over the region bounded by x = 0, x = 2, y = 0, y = 2.

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7. Find the limits in the integral where R is bounded by y = x2, x = 1 and x-axis.

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8. Find the value of

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9. Evaluate

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10. Change the order of integration in

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11. Evaluate

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12. Evaluate 

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13. Evaluate 

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14. Evaluate 

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15. Change the order of integration in

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After changing the order, first we have to integrate w.r.to x

16. Change the order of integration in

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17. Change the order of integration in

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18. Transform into polar coordinates.

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19. Find the area bounded by y = x and y = x2.

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20. Change the order of integration in

21. Evaluate

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22. Evaluate

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23. Evaluate

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24. Evaluate

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25. Evaluate

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26. Evaluate

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27. Change the order of integration in

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28. Change the order of integration,

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29. Find the area of a circle of radius 'a' by double integration in polar coordinates.

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30. Find over the region bounded by x ≥ 0, y ≥ 0, x + y ≤ 1.

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31. Evaluate

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32. Express in polar coordinates.

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33. Find the limits of integration in the double integral where R is in the first quadrant and bounded by x = 1, y = 0, y2 = 4x. 

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34. Write down the double integral, to find the area between the circles r = 2sinθ and r = 4sinθ.

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35. Change the order of integration

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When we change the order of integration, we have to integrate first w.r.to x. When we take strip parallel to x-axis, the integral splits into two parts, as there are two types of horizontal strips.

36. Evaluate

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37. Evaluate where C is the path y = x from (0, 0) to (1, 1).

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38. Transform into polar coordinates the integral

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The region of integration is the shade area OAB.

Transforming to polar coordinates x = rcosθ, y = rsinθ

when x = 0, r = 0

when x = a, a = rcosθ ⇒ r = a secθ

39. Why do we change the order of integration in multiple integral? Justify your answer with an example.

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In a double integral if the inner integral is difficult to evaluate we perform the technique of changing the order of the integral.

Here integration w.r.to y is impossible. So we change the order of integration. Changing the order of integration, first we integrate w.r.to x.

Then x varies from 0 to y

and y varies from 0 to ∞

40. Sketch roughly the region of integration for the following double integral

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41. Express the volume bounded by x ≥ 0, y ≥ 0, z≥ 0 and x2 + y2 + z2 ≤ 1 in triple integration.

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42. Evaluate

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43. Change the order integration in

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Changing the order of integration y varies from x to 1 and x varies from 0 to 1.

44. Evaluate

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45. Evaluate

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46. Evaluate

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47. Evaluate

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48. Sketch the region of integration in

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The region of integration is the sketched region as in figure