MATH 20C Lecture 9

13.1 Vector-Valued Integration

Definite Integral

Let
Define

Antiderivative

An antiderivative of is a vector-valued function s.t. .

Indefinite Integral

The indefinite integral is the family of all antiderivatives of .

Fundamental Theorem of Calculus for Vector-Valued Functions

1. If is continuous on and is an antiderivative of
   Then
2. is continuous on an open interval
   Let . Then,
   is an antiderivative of
   i.e.

13.3 Arc Length

Velocity vector

position vector
Then

1. is the velocity vector
2. is the speed function

Let $C$ be the curve parameterized by $\vec{r}(t)=⟨x(t),y(t),z(t)⟩$ for $a\le t\le b$

Then the arc length of over is = distance traveled