Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due
MAT 237: Multivariable Calculus with Proofs
Assignment #9
Due July 29, 11:59 PM EST
1. Reduce the calculation of each of the following integrals to iterated integrals which are well-defined. Once at this point you do not have to actually calculate the integral. You may assume that each function is integrable on the given set and that the conditions of Fubini’s theorem are satisfied. If you use change of variables, you don’t have to justify your choices therein. That being said, show your work and don’t just write down your final answer.
a),S exyzdV where S is the set in R3 bounded by z = x2 + y2 and z = 1.
b) ,S xydV where S is the set in R3 bounded by the x-axis, x = 1 − 4/y2, x = 4/y2 − 1, and x = 4 − 16/y2.
c),S x+y/1 dV where S is the set in the first quadrant of R2 bounded by x + y = 1 and x + y = 4.
d),S (x-y)5/(x+y)4dV where S is the set in R2 defined by the equations −1−y ≤ x ≤ 1−y andy+1 ≤ x ≤ y+3.
e),B1(0(⃗)) dV where B1(0(⃗)) is the closed unit ball in R4 .
2. For k ∈ N>0, let Ak = [1 + 
, k] × [1 + 
, k] and Bk = [1 + 
, k] × [1 + 
, 4k] be sets in R2 . Calculate the integral of the function
on Ak using a change of variables which makes use of the fact that Ak is symmetric around y = x and f is antisymmetric around the same line. Make sure to check the conditions for a change of variables in this problem. Then show that the integral of f on Bk tends to ∞ as k → ∞ , using whichever methods you wish. You may assume that f is integrable on these sets and that the conditions of Fubini’s theorem are satisfied for both calculations.
Note: This problem shows that some limits of integrals may depend on the chosen exhaustion, so the improper integral is not always well-defined.
3. Determine which of the following integrals converge. Determine which of them converge absolutely. Show your work. You don’t have to calculate any integrals here.
f) ,R3\0 dV , where ρ is a continuous real-valued function that is only nonzero in some bounded set, and ⃗a is a fixed point in R3 .
g) ,R2\0 ln∥⃗x∥ρ(⃗a − ⃗x)dV where ρ is a continuous real-valued function that is only nonzero in some bounded set and ⃗a is a fixed point in R2 .