MATH 1260 - Quiz 9
Solution
(1) Calculate the volume of the region bounded by the four planes x = 2, y = 3, z = 0, and
x − y − z + 3 = 0 by setting up an integral in the order dy dx dz.
Z 2Z
0
2
z
Z
x−z+3
1 dy dx dz =
3
=
Z 2Z
Z0 2 Zz 2
Z 2Z
=
=
=
=
=
→
((x − z + 3) − 3) dx dz
2
(x − z) dx dz
2 !
Z 2 2
x
− xz
dz
2
0
z
2
Z 2
z
2
dz
−z
(2 − 2z) −
2
0
Z 2
z2
dz
2 − 2z +
2
0
2
z3
2
2z − z +
6 0
8
4−4+
6
4
3
0
=
[y]x−z+3
dx dz
3
z
0
=
2
→
z
→
→
(2) Evaluate the line integral of F (x, y, z) = x2 i − xy j + k along the circle of radius 1, center at
the origin and lying in the yz plane.
The circle of radius 1 with center at the origin and lying in the yz plane is parametrized by
→
r (t) = h0, cos t, sin ti with 0 ≤ t ≤ 2π. The line integral is
Z 2π
Z 2π
h0, 0, 1i · h0, − sin t, cos ti dt =
cos t = sin t|2π
0 = 0 − 0 = 0.
0
0
→
→
(Extra-credit) Show that if a particle is moved along a closed curve (that is, r (a) = r (b)), then
the work done on it by the vector field
→
→
→
→
1
(x
i
+
y
j
+
z
k)
F (x, y, z) = − 2
(x + y 2 + z 2 )3/2
is zero.
→
We have done this in class. It is because F (x, y, z) is the gradient of
f (x, y, z) =
(x2
1
1
.
+ + z 2 )1/2
y2