ENGR 233 MIDTERM SOLUTIONS - CLASS TEST ~ CONCORDIA UNIVERSITY

ENGR 233 MIDTERM SOLUTIONS - CLASS TEST ~ CONCORDIA
UNIVERSITY


Engr 233, Section R; Class test, Friday, March 22, 2024
Instructor A. Kokotov

Time: 60 min

Answer all questions. You may use your own written notes. No electronic devices!


1. (10 points) Let A = (0, 1, 1) and B = (1, 2, 3). Prove that the follow-
ing line integral is path independent, and compute it either choosing an
appropriate contour of integration or using the potential function:
∫ B
(2xy2 + 2xz 2)dx + (2yx2 + z)dy + (2x2 z + y)dz
A

2. (10 points) Change the order of integration (i. e. put the limits of
integration at the right hand side)

∫ √ ∫ √ ! ∫ ∫ ∫ ∫
1/ 2 1−x2

√ f (x, y) dy dx = f (x, y) dx dy+ f (x, y) dx dy
−1/ 2 |x|

3. (10 points)
Let the domain Ω be defined by the inequalities
x2 + y2 ≥ 1; x2 + y2 ≤ 4; y ≥ x; y≥0

Compute the double integral

∫∫
Ω
y(x2 + y2)3/2 dx dy
4. (10 points) Compute the line integral
I
xdy − y dx
Γ
where Γ is the counterclockwise oriented boundary of the domain defined
by the inequalities