Calculus II, Fall 2013

Quiz 5

May 30, 2013

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1. Evaluate

∫_{0}^{1} ∫_{y/2}^{y/2+1} y^{3}(2x − y)e^{(2x−y)^{2}} dx dy

2. Find the work done by F (x, y, z) = xyi + yzj + x^{2}k over the line segment joining straightly from (1, 1, 1) to (0, 0, 0). Does the answer remain the same if we change it to a different path joining from (1, 1, 1) to (0, 0, 0)?

3. Is the vector field F (x, y, z) = (z + y)i + zj + (y + x)k conservative? If so, find the potential. If not, explain why.

4. State Green’s Theorem for the vector field F (x, y) = xyi + y^{2}j on the unit disk {x^{2} + y^{2} < 1}. You can choose either one of the two forms. Then evaluate the corresponding double integral and line integral and check they are the same.

Calculus II, Fall 2013

Quiz 5

May 30, 2013

Show all details.

1. Evaluate

∫_{0}^{1} ∫_{y/2}^{y/2+1} y^{3}(2x − y)e^{(2x−y)^{2}} dx dy

2. Find the work done by F (x, y, z) = xyi + yzj + x^{2}k over the line segment joining straightly from (1, 1, 1) to (0, 0, 0). Does the answer remain the same if we change it to a different path joining from (1, 1, 1) to (0, 0, 0)?

3. Is the vector field F (x, y, z) = (z + y)i + zj + (y + x)k conservative? If so, find the potential. If not, explain why.

4. State Green’s Theorem for the vector field F (x, y) = xyi + y^{2}j on the unit disk {x^{2} + y^{2} < 1}. You can choose either one of the two forms. Then evaluate the corresponding double integral and line integral and check they are the same.