9.5 Integrating Vector-Valued Functions

• The final integrated vector function $\mathbf{r}(t)$ includes constants found using initial conditions.
• What is the final form of the $x$-component after integrating \mathbf{v}(t) = \langle 3t^{2}, 2t \rangle</latex> with $\mathbf{r}(0) =$$\langle 1, 2 \rangle$?


  $t^{3} +$$1$
• What is a vector-valued function?

  Maps real number to vector
• The component functions of a vector-valued function are f(t), $g(t)$, and $h(t)$.
• What is the general form of a vector-valued function?

  $\mathbf{r}(t) =$$\langle f(t), g(t), h(t) \rangle$
• Vector-valued functions extend the concept of parametric equations to vector form.
• A vector-valued function maps a real number to a vector in two or three dimensions.
• What are the component functions of a vector-valued function?

  $f(t), g(t), h(t)$
• To differentiate a vector-valued function, you differentiate each component function separately.
• What does the derivative $\mathbf{r}'(t)$ of a vector-valued function represent?


  Tangent to the curve
• How do you differentiate a vector-valued function $\mathbf{r}(t) =$$\langle f(t), g(t), h(t) \rangle$?


  Differentiate each component separately
• Differentiating a vector-valued function involves differentiating each component separately.
• The tangent vector $\mathbf{r}'(t)$ represents the slope of the curve at $t$.
• The indefinite integral of a vector-valued function is found by integrating each component separately.
• What must be included in each component integral when finding the indefinite integral of a vector-valued function?

  Constant of integration
• Each component integral in the indefinite integral of a vector-valued function includes a constant of integration.
• Match the component with its indefinite integral:

  $f(t)$ ↔️ $\int f(t) \, dt +$$C_{1}$

  $g(t)$ ↔️ $\int g(t) \, dt +$$C_{2}$

  $h(t)$ ↔️ $\int h(t) \, dt +$$C_{3}$
• A vector-valued function maps a real number $t$ to a vector in two or three dimensions.
• What are the component functions of a vector-valued function called?

  $f(t), g(t), h(t)$
• The vector $\mathbf{r}(t)$ represents the position of the curve at parameter $t$.
• What does the component function $f(t)$ represent in a vector-valued function?


  x-coordinate
• The derivative $\mathbf{r}'(t)$ of a vector-valued function is the vector representing the tangent to the curve.
• Arrange the concepts in the correct order to describe a vector-valued function and its differentiation:

  1️⃣ Vector-valued Function $\mathbf{r}(t)$

  2️⃣ Component Functions $f(t), g(t), h(t)$

  3️⃣ Differentiation $\mathbf{r}'(t)$

  4️⃣ Tangent Vector $\mathbf{r}'(t)$
• How do you differentiate a vector-valued function $\mathbf{r}(t) =$$\langle f(t), g(t), h(t) \rangle$?


  Differentiate each component separately
• The derivative of a vector-valued function $\mathbf{r}(t) =$$\langle f(t), g(t), h(t) \rangle$ is given by \mathbf{r}'(t)
• The derivative $\mathbf{r}'(t)$ represents the tangent to the curve at point $t$.
• What does a vector-valued function map $t$ to?


  \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle</latex>
• A vector-valued function $\mathbf{r}(t)$ consists of three component functions $f(t), g(t)$, and $h(t)$.
• Match the concept with its mathematical expression:

  Differentiation ↔️ $\mathbf{r}'(t) =$$\langle f'(t), g'(t), h'(t) \rangle$

  Tangent Vector ↔️ $\mathbf{r}'(t)$

  Vector-valued Function ↔️ $\mathbf{r}(t) =$$\langle f(t), g(t), h(t) \rangle$
• Steps to find the indefinite integral of a vector-valued function

  1️⃣ Integrate each component separately

  2️⃣ Add a constant of integration to each integral

  3️⃣ Combine the results into a new vector-valued function
• Each indefinite integral of a vector-valued function includes a constant of integration.
• What is the indefinite integral of the component f(t)</latex> in $\mathbf{r}(t)$?


  $\int f(t) \, dt +$$C_{1}$
• Match the component with its indefinite integral:

  $f(t)$ ↔️ $\int f(t) \, dt +$$C_{1}$

  $g(t)$ ↔️ $\int g(t) \, dt +$$C_{2}$

  $h(t)$ ↔️ $\int h(t) \, dt +$$C_{3}$
• How do you integrate a vector-valued function $\mathbf{r}(t) =$$\langle f(t), g(t), h(t) \rangle$?


  Integrate each component separately
• When integrating a vector-valued function, each component requires its own constant of integration.
• What is the indefinite integral of $2t$?


  $t^{2} +$$C_{1}$
• How do you find the definite integral of a vector-valued function $\mathbf{r}(t) =$$\langle f(t), g(t), h(t) \rangle$ from $a$ to $b$?


  Integrate each component from $a$ to $b$
• The definite integral of $\langle 2t, 3t^{2}, 4t^{3} \rangle$ from 0 to 1 is $\langle 1, 1, 1 \rangle$.
• What does the integral of a vector-valued velocity function represent?

  Displacement
• What is the displacement of a particle moving with velocity $\mathbf{v}(t) =$$\langle 2t, 3t^{2} \rangle$ from $t =$$0$ to $t =$$2$?


  $\langle 4, 8 \rangle$