Integrals of Vector Valued Functions

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Integrals of Vector-Valued Functions

Based on “Integrals of Vector-Valued Functions” in section 2.2 of the textbook.

Key Ideas

The integral (either definite or indefinite) of a vector-valued function is a vector of integrals of the function’s components.

Warm-Up Examples

Find the antiderivative of ⟨ t² - 2t, cos t, sin (3t) ⟩

Use antiderivative rules and integration methods (in this case, u-substitution to integrate sin(3t) — all the methods of integration that you know are usable in vector-valued functions too) to integrate each component.

How about the definite integral of ⟨ t² - 2t, cos t, sin (3t) ⟩ from t = 0 to t = π?

Evaluate the antiderivative from above between the bounds 0 and π. Notice that you can use the idea of cosine being the x coordinate of a point on a unit circle to work out that cos(3π) must be -1.

An Initial Value Problem

Imagine a drone aircraft subject to acceleration ⟨ t/5, -1, cos t ⟩ (acceleration is in m/sec², t represents time in seconds). If the drone’s velocity when t = 0 is ⟨ 0, 3, -1 ⟩ (m/sec), what is its velocity as a function of time? (Hint: acceleration is the derivative of velocity).

Start by integrating the acceleration function to find the velocity function:

That velocity function has constants of integration in it. To figure out what they are, use the fact that plugging t = 0 into the velocity function needs to give you the known v(0); that fact lets you solve for the constants of integration (which in this case happen to equal v(0), but it doesn’t have to turn out that way):

Integration in Mathematica

Use the built-in Integrate function, which works very much like D, i.e., it takes a function to integrate and the variable to integrate with respect to as its arguments. To compute a definite integral, include bounds with the variable.

I demonstrated these uses. You can download the notebook from Canvas in order to see the demonstration and experiment with them.

Next

If vector-valued functions describe curves in space, it might be interesting to talk about the lengths of those curves (e.g., how far you’d travel if you walked along one, how much wire you’d need to make one, etc.)

For example, how long is the helix r(t) = ⟨ cos (πt), sin (πt), t ⟩ if t ranges from 0 to 4?

You can work out a formula for arc length by starting with a single very small (i.e., infinitesimally small) segment of a curve, and using the distance formula to calculate its length. Then integrating (i.e., adding up) all those lengths over the entire curve gives you its length:

To learn more, please read “Arc Length for Vector Functions” and “Arc-Length Parameterization” in section 2.3 of the textbook.

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