SUNY Geneseo Department of Mathematics

Integrals of Vector Valued Functions

Thursday, February 20

Math 223 01
Spring 2020
Prof. Doug Baldwin

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Previous Lecture

Misc

Hour Exam 1

Next Thursday (Feb. 27).

Covers material since the beginning of the semester (e.g., 3D coordinates, lines/planes/surfaces, vectors, vector-valued functions, their derivatives and integrals, etc.)

3 - 5 short-answer questions, similar to mid-difficulty questions from problem sets.

You’ll have the whole class period.

Open book, notes, calculator, computer as a reference or calculator. But closed person.

Questions?

Proving the sum limit law for problem set? A good way to do many proofs about limits of vector-valued functions is to move the limits inside the vector, use what you already know about limits of scalar functions, and then move the resulting limits back out of the vector.

Integrals of Vector-Valued Functions

Last subsection of section 12.2 in the textbook.

Key Ideas

The antiderivative of a vector-valued function is a vector of antiderivatives of components. Constants of integration form a constant vector of integration.

The definite integral of a vector-valued function is the vector of definite integrals of components.

The Fundamental Theorem of Calculus applies to vector-valued functions.

Example

Find an antiderivative and a definite integral. The key is to find antiderivatives and integrals of the components. And to remember that all the techniques of integration you know for scalar functions can be applied to the component functions.

Integrating components in a vector-valued function for antiderivative and definite integral

Application

Once when I was lighting up an outdoor grill a fly flew over it and got killed by the heat. What trajectory did the fly then follow?

Fly passing over grill with acceleration and initial velocity and position

Start by integrating acceleration to get velocity, using the initial velocity to solve for the constant of integration:

Antiderivative of acceleration is velocity, with constant set to give known velocity when t is 0

The similarly integrate velocity to get position:

Integral of velocity is position, with constant set to give known initial position

Next

Arc length of vector-valued functions.

Read “Arc Length for Vector Functions” and “Arc-Length Parameterization” in section 12.4.

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