Here is an example of setting up line integrals for a piecewise smooth path. It should be noted that the two paths involved here are some of the key examples I highlight in my classroom.
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Integration by Substitution, a more interesting case
Critical Numbers 7
Critical Numbers 6
Critical Numbers 5
Critical Numbers 4
Critical Numbers 3
Critical Numbers 2
Critical Numbers 1
Vector Parallel to a Plane
Curve Sketching Summary Example
Simplifying the Quotient Rule
Line Integrals 9
Triple Integrals 11 b
Triple Integrals 11 a
Lagrange Example 3
Curvature Examples part 2
Arclength Examples part 2
Vector Function Proof
Quick Review of Some Key Functions
Surface Area 2
Mass Center of Mass 2
Vectors in 3 Dimensions part 2
Vectors in 3-D part 1
Introduction to the Equation of a Sphere