Line integrals - University of Alberta

MATH 209—
Calculus,
III
Volker Runde
Line integrals
in R 2
Types of line
integrals
Line integrals
in R 3
Line integrals
of vector fields
Examples, IV
Example
Let C consist of the paraboloa y = x 2 from (0, 0) to (1, 1)
followed by the vertical line segment from (1, 1) to (1, 2).
Find �
C 2x ds.
The curve C is not smooth, but piecewise smooth, i.e., of the
form C = C1 ⊕ C2 with C1 and C2 smooth.
C1: the parabola y = x 2 from (0, 0) to (1, 1):
x = t, y = t 2 , t ∈ [0, 1].
C2: the line segment from (1, 1) to (1, 2):
r(t) = 〈1, 1〉 + t(〈1, 2〉 − 〈1, 1〉) = 〈1, 1 + t〉
for t ∈ [0, 1], i.e., x = 1, y = 1 + t, and t ∈ [0, 1].