Integrals in R2 and R3
Bhavin Mahendra
10th July 2023
1
Exercise 1.
4 points
Let A = {(x, y, z) ∈ R3 : x > 0, y > 0, 1 + x + y < z < 1 − 2x} be a region. Determine an expression for
the following iterated integrals and calculate the volume of A:
1.
RRR
2.
RRR
3.
RRR
A
A
A
1dydzdx
1dxdzdy
1dxdydz
2
Exercise 2.
4 points
Let B = {(x, y, z) ∈ R3 : x > 0, y > 0, z > 0, x < y, y < z9 < x2 + y 2 + z 2 < 25} be a region. Using the
right change of variables, calculate the volume of B.
3
Exercise 3.
8 points
Let F : R2 →
− R2 be a vector field.
F (x, y) =
Exercise 3.1.
(x + 1)2 + (y − 1)2 + 2
(x + 1)2 + (y − 1)2
2x + xy
,
,
2
2
x+y
3(x + 1) + 3(y − 1)
2x + 5y 2
4 points
Calculate the line integral of F around a circumference of center (-1,1) and radius 4 using a parametrization.
Exercise 3.2.
4 points
Evaluate if F is a closed path. Calculate the line integral of F around a circumference of center (-1,1)
and radius 4 using Green’s Theorem.
4
Exercise 4.
4 points
Let F : R3 →
− R3 be a conservative vector field.
F (x, y, z) =
y2 z3 +
−sin(πx)
sin(πx)
, 2xyz 3 +
, 3xy 2 z 2
−
e xy
ex y
Determine the potencial function of F and calculate the line integral of F.
5