Consider the double integral 0int20intradic1-x-12 xy3x2 y2

Question

1. Consider the double integral ₀∫²₀∫^{√(1-(x-1)2)} xy³/(x²+ y²) dydx.

a. Explain why this integral is inconsistent with the definition of the double integral.

b. Explain why Fubini's Theorem does not apply to this integral as written.  

c. Describe the behavior of f(x, y) = (xy³/x²+y²)as (x, y) → (0, 0).

d. Treat the integral as an improper integral and use correct notation for improper integrals to evaluate the double integral. Be sure to use correct notation and justify every step of your work.

 2. Evaluate the integral ∫∫∫_D e^{-√(x2+y2+z2)} dV where D is all of R³. Note that this is an improper integral, so you should use correct notation and thoroughly justify your steps.

3. Use an appropriate change of variables to evaluate the integral ∫∫_R (2x+2y)e^x2-y2 dA over the rectangle with vertices (0, 0), (5/2, 1/2), (3/2, 3/2), and (1, -1).