MATH 251 – LECTURE 24
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 14.1–2
webAssign: 14.1–2, due 4/4 11:55 p.m.
(No webAssignments need to be handed in.)
Next week: 14.3–4
webAssign: 14.3–4, opens 4/4 12 a.m.
Help Sessions:
Sun–Thu 6–8 p.m. in BLOC 149
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Curve integrals
Let f (x, y) be a function of two variables. We had defined the curve integral
Z
n
X
f (x, y) ds = lim
f (x∗i , yi∗)∆si.
C
|P |→0
i=1
Curve integrals
Now define the curve integrals with respect to x and y:
Z
n
X
f (x∗i , yi∗)∆xi.
f (x, y) dx = lim
C
|P |→0
Z
f (x, y) dy = lim
C
|P |→0
i=1
n
X
i=1
f (x∗i , yi∗)∆yi.
Curve integrals
And in general, we write
Z
Z
P (x, y) dx + Q(x, y) dy =
C
Z
P (x, y) dx +
C
Q(x, y) dy
C
Exercise 1. Let C be the line segment from (0, 0) to (2, 2). Evaluate the line integral
Z
xy dx + (x − y) dy.
C
Curve integrals
Exercise 2. Let C consist of the arc of the circle x2 + y 2 = 1 from (1, 0) to (0, 1) and the line segment from
(0, 1) to (4, 3). Evaluate the line integral
Z
y dx − x dy.
C
Curve integrals
Exercise 2. Let C consist of the arc of the circle x2 + y 2 = 1 from (1, 0) to (0, 1) and the line segment from
(0, 1) to (4, 3). Evaluate the line integral
Z
y dx − x dy.
C
Curve integrals
In three dimensions
Z
P (x, y, z) dx + Q(x, y, z) dy + R(x, y, z) dz
C
Z b
(P (x(t), y(t), z(t)) x0t + Q(x(t), y(t), z(t)) yt0 + R(x(t), y(y), z(y)) zt0 ) dt.
=
a
Curve integrals
Let F (x, y) = hP (x, y), Q(x, y)i be a vector field, and let dr = hdx, dyi. Then we can form the integral
Z
Z
F (x, y) · dr =
P (x, y) dx + Q(x, y) dy.
C
C
Curve integrals
The work done by the vector field F (x, y) = hP (x, y), Q(x, y)i on a particle that moves along the curve C is
given by the integral
Z
F (x, y) · dr.
C
Exercise 3. Find the work done by the field F (x, y) = hx sin(y), yi on a particle that moves along the parabola
y = x2 from (−2, 4) to (1, 1).
Curve integrals
Exercise 4. Let F1 = hy, xi and let F2 = h2y, xi. Compute the Curve integrals of F1 and F2 from P1(1, 0) to
P2(0, 1) when a) C1 is the line segment from P1 to P2, and b) C2 is the arc in the circle x2 + y 2 = 1.
Curve integrals
Exercise 4. Let F1 = hy, xi and let F2 = h2y, xi. Compute the Curve integrals of F1 and F2 from P1(1, 0) to
P2(0, 1) when a) C1 is the line segment from P1 to P2, and b) C2 is the arc in the circle x2 + y 2 = 1.