Practice questions Test 4
Instructor: Dr. Sarobidy Razafimahatratra
MATH 2004F/2008F
Winter 2026
Z 1Z π
2
1) Evaluate the integral
0
4 cos3 (x) dx dy by interchanging the order of integration.
arcsin(y)
1
2) Use the change of variables x =
1
(u − v) and y = 16 (u + 5v) to compute the integral
6
where R is the region enclosed by the parallelogram determined by
(
−5x ≤ y ≤ −5x + 1
x − 1 ≤ y ≤ x + 2.
2
Z Z
(x + y)dA,
R
Z Z
Z Z
F(x, y, f (x, y))·(−fx , −fy , 1) dA of the vector field F = (−x, −y, z)
p
through S , where S is the cone with equation z = f (x, y) = x2 + y 2 between the planes z = 3 and
z = 5.
3) Compute the flux
F·dS =
S
R
3
4) Green’s theorem states that if F(x, y) = (P (x, y), Q(x, y)) is a vector field in R2 , C is a positively oriented
piecewise smooth closed curve, and R is the region bounded by C , then
Z
Z Z ∂Q ∂P
F · dr =
−
dA.
∂x
∂y
C
R
Z
Use Green’s theorem to compute the line integral
F·dr, where C is the ellipse with equation x2 +5y 2 = 3
and F = (P, Q) = (xy, x2 ).
C
4
Z
5) Use Green’s theorem to compute the line integral
2
2
ellipse xa2 + yb2 = 1, where a, b > 0.
C
5
xdy, where C is the positively oriented curve on the