By now, you should have a good understanding of vector functions. It’s alright if you don’t, for a refresher you can read our 9.4 study guide which talks about defining and differentiating vector-valued functions. This will help you understand this study guide and new concept easier!

Integrating these vector-valued functions allows us to go backward when relating the movement of a particle to its equation. For example, let's say we have a vector-valued function that gives the acceleration of a particle moving in space. A question asks us to determine the velocity of said particle. This means we need ✨integration✨

💭 Integrating Vector-Valued Functions

It’s good to first recognize the format of your function. It is either written in two or three dimensions; we want to know which one so that we are not confused about which terms should be paid attention to when integrating.

A two-dimension vector function might look like this:

r(t)=f(t)i+g(t)j

or like this:

r(t)=⟨f(t),g(t)⟩

These both mean the same thing! 🪄

❓ How do we integrate?

Luckily, we integrate each component of a vector function separately. No need to make things more complicated than they need to be! You look at each component as if it was its own problem. Let’s go through an example just to see how the College Board might ask a question like this.

✏️ Integrating Vector-Valued Functions Walkthrough

We are given:

r(t)=⟨t^2,\sin(t),e^t⟩

We know we are going to need to integrate this vector-valued function, so let’s re-write this including integrals on each component.

\int{r(t)}dt=⟨∫t^2dt,∫\sin(t)dt,∫e^tdt⟩

You can solve each of these integrals separately. If you need help with integrals, check out our guide 6.14 Selecting Techniques for Antidifferentiation. Although it says for AB, it will apply to this!

Our final answer will be:

R(t)=⟨\frac13t^3+C1,−\cos(t)+C2,e^t+C3⟩

Please note that C1, C2, and C3 just represent different possible constants and I added them for clarity. For these types of problems, you will typically get definite integrals so that there is less ambiguity. Follow your typical steps for solving these problems!

📝 Additional Practice Problem

Let’s try a word problem now (note this is a calculator-okay question). This problem (#9) is from the flipped math calculus website. All credit to them.

At time 𝑡 ≥ 0, a particle moving in the 𝑥𝑦-plane has a velocity vector given by.

v(t)=⟨3t^2,3⟩

If the particle is at point (1,2) at time 𝑡 = 0, how far is the particle from the origin at time 𝑡 = 2?

Try the problem on your own first! If you are stuck read through our explanation below!

🏁 Solution to Practice Problem

Step one is integrating the function. As we’ve discussed, you will integrate each of the components of the vector function separately. Before applying conditions, this is what you should have for your integrated vector function:

x(t)=∫3t^2dt=t^3+Cx

y(t)=∫3dt=3t+Cy

The Cx and Cy constants are just placeholders. We are going to figure out those numbers now:

Since we are given the position for t=0, we can use that information to work backward and find our constants.

x(t):1=0^3+Cx\therefore Cx=1

y(t):2=3(0)+Cy\therefore Cy=2

Now that we know the values for our constants let's put those into our integrated vector functions:

⟨t^3+1,3t+2 ⟩

Now we’re done with the hard parts! From here on it’s just simple math to finish off the problem. We will plug in t=2 to our equation to find the position of the particle at that point. This gives us

(9,8)

Finally, we’ll use the handy distance formula to find how far this point is from the origin. Since this would be a question in the calculator section, you’ll want to plug this into your calculator and give a number rounded to three decimal places rather than leave it in radical form.

\sqrt{(9-0)^2+(8-0)^2}

You should get 12.04 units as your final answer!

Why integrate vector-valued functions?

With these types of problems, you want to have a good understanding of what you can do with the functions. Vector functions often tell you the movement or position of a particle. This means the integrated function can give the total area moved over time or similar. Don’t worry too much about the nitty-gritty, but practice recognizing graphs for regular functions and matching them to their differentiated or integrated functions. You can get practice with this in our guide 9.6! Good luck and you’ve got this. 😁

Image Courtesy of Giphy

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Unit 9

9.5 Integrating Vector-Valued Functions

2 min read•june 18, 2024