Calculus II, Spring 2014

Quiz 5

May 29, 2014 Show all details.

1. Evaluate ∫_{−1}^{0} ∫_{−\sqrt{1−x^{2}}}^{0} \frac{2}{1 + \sqrt{x^{2} + y^{2}}} dy dx

2. Evaluate ∫_{0}^{4} ∫_{0}^{1} ∫_{2y}^{2} \frac{2 cos(x^{2})}{\sqrt{z}} dx dy dz

3. Express the volume of D = \{(x, y, z), x^{2} + y^{2} < 1, 0 < z < \sqrt{4 − x^{2} − y^{2}}\} as a triple integral in cylindrical and spherical coordinates, respectively. Need not evaluate them.

4. Evaluate ∫_{1}^{2} ∫_{1/y}^{y} \sqrt{\frac{y}{x}}e^{\sqrt{xy}} dx dy

5. Let F (x, y) = (y, x) and C = \{(x, y), \frac{x^{2}}{4} +y^{2} = 1\}. Evaluate ∫_{C} F ·T ds and ∫_{C} F ·nds where T is the counter-clockwise unit tangent and n the outward unit normal on C.

Calculus II, Spring 2014

Quiz 5

May 29, 2014 Show all details.

1. Evaluate ∫_{−1}^{0} ∫_{−\sqrt{1−x^{2}}}^{0} \frac{2}{1 + \sqrt{x^{2} + y^{2}}} dy dx

2. Evaluate ∫_{0}^{4} ∫_{0}^{1} ∫_{2y}^{2} \frac{2 cos(x^{2})}{\sqrt{z}} dx dy dz

3. Express the volume of D = \{(x, y, z), x^{2} + y^{2} < 1, 0 < z < \sqrt{4 − x^{2} − y^{2}}\} as a triple integral in cylindrical and spherical coordinates, respectively. Need not evaluate them.

4. Evaluate ∫_{1}^{2} ∫_{1/y}^{y} \sqrt{\frac{y}{x}}e^{\sqrt{xy}} dx dy

5. Let F (x, y) = (y, x) and C = \{(x, y), \frac{x^{2}}{4} +y^{2} = 1\}. Evaluate ∫_{C} F ·T ds and ∫_{C} F ·nds where T is the counter-clockwise unit tangent and n the outward unit normal on C.