Calculus II, Spring 2016

Brief answers to Quiz 5

May 26, 2016

1. (20 pts) Rewrite ∫_{−1}^{1} ∫_{x^{2}}^{1} ∫_{0}^{1−y}dzdydx in the order dydzdx and dxdydz, respectively.

Need not evaluate it. Answer.

∫_{−1}^{1} ∫_{0}^{1−x^{2}} ∫_{x^{2}}^{1−z} dydzdx (10 pts)

∫_{0}^{1} ∫_{0}^{1−z} ∫_{−\sqrt{y}}^{√\overline{y}} dxdydz (10 pts)

2. (20 pts) Let D be the region bounded below by z = 0, above by x^{2} + y^{2} + z^{2} = 4 and the sides by x^{2} + y^{2} = 1. Express the volume of D in terms of dzdrdθ and dρdϕdθ, respectively. Need not evaluate it. Answer.

∫_{0}^{2π} ∫_{0}^{1} ∫_{0}^{√\overline{4−r^{2}}} r dzdrdθ. (8 pts)
∫_{0}^{2π} ∫_{0}^{π/6} ∫_{0}^{2} ρ^{2} sin ϕ dρdϕdθ + ∫_{0}^{2π} ∫_{π/6}^{π/2} ∫_{0}^{csc ϕ} ρ^{2} sin ϕ dρdϕdθ. (12 pts)

3. (20 pts) Evaluate ∫ ∫_{R} \sqrt{\frac{y}{x}} + \sqrt{xy}dxdy where R is the region bounded by xy =

1, xy = 4, y = x and y = 3x in the first quadrant. Answer. Let x = u/v, y = uv. Then |\frac{∂(x,y)}{∂(u,v)}| = 2u/v. And,

∫_{1}^{√\overline{3}} ∫_{1}^{2}(u + v)\frac{2u}{v} dudv (16 pts) = \frac{7}{3} ln 3 + 3\sqrt{3} − 3. (4 pts)

4. (20 pts) Evaluate ∫_{C} \frac{x^{2}}{y^{\frac{4}{3}}} ds where x(t) = t^{2}, y(t) = t^{3}, 1 ≤ t ≤ 2.

Answer.

∫_{C} \frac{x^{2}}{y^{\frac{4}{3}}} ds = ∫_{1}^{2} \frac{t^{4}}{t^{4}} \sqrt{4t^{2} + 9t^{4}} dt (16 pts) = \frac{1}{27}(40\sqrt{40} − 13\sqrt{13}). (4 pts)

5. (20 pts) Evaluate ∫_{C} F · T ds where F = (x^{2}, −y) and C is the curve x = y^{2} from

(4, 2) to (1, −1). Answer. Take r(t) = ((1 − t)^{2}, 1 − t), −1 ≤ t ≤ 2.

∫_{C} F · T ds = ∫_{−1}^{2}((1 − t)^{4}, −(1 − t)) · (−2(1 − t), −1) (16 pts)dt = −\frac{39}{2}. (4 pts)