Vector calculus-three dimensional space

Question

Problems:

Vector Calculus: Three Dimensional Space

(1) Consider the curve a→(t) = ti^ˆ+ t2j^ˆ₊ t3k^ˆfor -1<t<1 and denote by I its path.
Compute the line integral over r of the vector field
                                             F^→ = y²i+2xyj.

(2) Consider the following four curves:

             a^→A(t) = sin tiˆ+ cos44tjˆ+ sin5t kˆ                   - ∏/2 < t < ∏/2
             a^→B (t) = (t+1)iˆ+ (t+1)2jˆ+(t+1)3kˆ                         -1 < t < 1
             a^→C (t) = t⁴iˆ - t¹²jˆ- t²kˆ                                            -1< t < 1
             a^→D (t) = (4/1+t - 3) (iˆ + kˆ) + 1/1+t(1-t) jˆ               0 < t < 1.

Denote by τ_A,τ_B,τ_C and τ_D the corresponding paths. The integrals of the field F^→ defined above over τ_A,τ_B,τ_C and τ_D give the following values, listed in ascending order:
                                                -2, 0, 2, 32.
Associate to every curve the value of the corresponding line integral. Justify your answer.

Hint 1 : Use the result obtained in the first exercise.
Hint 2 : You do not need to compute any integral.

(3) Compute the double integral of the field ƒ= x+y/x over the region
                                         R= {xiˆ+ yjˆ∈ ℜ², 0 < x < 2,0 < 1/2 (y/x +1) < 1}.

Hint; The definition of the domain should suggest you a change of variables mapping R to a rectangle.

(4) Consider the following subsets of ℜ3:
                              T =  { r^→∈ℜ³, x² + 2y² + 3z = 1},
                              U = { r^→∈ℜ³, x² + 2y² + 3z² < 1},
                              V =  { r^→∈ℜ³, x² + 2y² + 3z² = 1^}
                              W = { r^→∈ℜ³, x² + 2y² + 3z^{2 < 1,} z = 0^}
                              X =  { r^→∈ℜ³, x² + 2y² + 3z² = 1, y = 0^}
                              Y =  { r^→∈ℜ³, x² + 2y² + 3z = 1, y = 0^}
                              Z =  { r^→∈ℜ³, x² + 2y² + 3z² > 1}

Which of these are:

(i) the path of a curve,
(ii) the path of a loop,
(iii) a two-dimensional (fiat) region,
(iv) a graph surface,
(v) the boundary of a domain,
(vi) the level set of a scalar field defined on R3,
(vii) an oriented surface,
(viii) a domain?
In this exercise you do not need to justify your answer.
MILL; Note that a given set may belong to more than one class.