Integrating Vector-Valued Functions - AP Calculus BC

1. Evaluate the integral $\textbf{r}(t) = \begin{pmatrix} t \\ t^2 \end{pmatrix}$ over $[1, 2]$.

   $\begin{pmatrix} \textstyle\frac{3}{2} \\ \textstyle\frac{7}{3} \end{pmatrix}$. Integrate $t$ to get $\frac{3}{2}$ and $t^2$ to get $\frac{7}{3}$ over $[1,2]$.

2. How do you represent the antiderivative of a vector function $\textbf{r}(t)$?

   $\textbf{R}(t) = \begin{pmatrix} F(t) \\ G(t) \\ H(t) \end{pmatrix}$. Each component is the antiderivative of the corresponding component in $\textbf{r}(t)$.

3. Calculate the integral $\textbf{r}(t) = \begin{pmatrix} 3t \\ 4 \end{pmatrix}$ over $[1, 3]$.

   $\begin{pmatrix} 9 \\ 8 \end{pmatrix}$. Integrate $3t$ to get $\frac{3}{2}(3^2-1^2)=9$ and $4$ to get $4(3-1)=8$.

4. Evaluate the integral of $\textbf{r}(t) = \begin{pmatrix} \text{sinh}(t) \\ \text{cosh}(t) \end{pmatrix}$ over $[0, 1]$.

   $\begin{pmatrix} \text{cosh}(1) - 1 \\ \text{sinh}(1) \end{pmatrix}$. Antiderivatives of hyperbolic sine and cosine functions.

5. State the relationship between definite and indefinite integrals for vector functions.

   $\text{Definite Integral} = \text{Indefinite Integral evaluated at bounds}$. Same principle as scalar functions: evaluate antiderivative at bounds.

6. State the formula for integrating a vector-valued function with respect to a parameter $u$.

   $\textbf{R}(u) = \begin{pmatrix} F(u) \\ G(u) \\ H(u) \end{pmatrix} + \textbf{C}$. General form with parameter $u$ instead of $t$.

7. In the context of vector-valued functions, what does $\textbf{C}$ represent?

   Constant vector of integration. Vector analog of the constant of integration in scalar calculus.

8. State the relationship between integration and differentiation for vector-valued functions.

   They are inverse operations. Differentiation and integration undo each other for vector functions.

9. What rule applies for integrating vector-valued functions with piecewise components?

   Integrate each piece separately. Apply integration rules to each piece within its domain.

10. Evaluate the integral $\textbf{r}(t) = \begin{pmatrix} t \\ t^2 \end{pmatrix}$ over $ [1, 2] $.

    $\begin{pmatrix} \textstyle\frac{3}{2} \\ \textstyle\frac{7}{3} \end{pmatrix}$. Integrate $t$ to get $\frac{3}{2}$ and $t^2$ to get $\frac{7}{3}$ over $ [1,2] $.

11. Express the integral $\mathbf{r}(t) = \begin{pmatrix} t^2 \\ \sin(t) \end{pmatrix}$ over $[0, 2]$ in component form.

    $\begin{pmatrix} \frac{8}{3} \\ 1 - \cos(2) \end{pmatrix}$. Integrate $t^2$ to get $\frac{8}{3}$ and $\sin(t)$ to get $1 - \cos(2)$.

12. How does the integral of a vector-valued function change with a change of variable?

    Use substitution method. Apply substitution rule component-wise with appropriate bounds transformation.

13. What is the derivative of the integral of a vector-valued function $\textbf{r}(t)$?

    $\textbf{r}(t)$. By the Fundamental Theorem of Calculus for vector functions.

14. Determine the integral of $\textbf{r}(t) = \begin{pmatrix} t \\ e^t \end{pmatrix}$ over $[0, 1]$.

    $\begin{pmatrix} \textstyle\frac{1}{2} \\ e - 1 \end{pmatrix}$. Integrate $t$ to $\frac{1}{2}$ and $e^t$ to $e-1$ over $[0,1]$.

15. What rule applies when integrating vector functions with respect to a parameter?

    Parameter integration rule. Integration is performed component-wise with respect to the parameter.

16. Calculate the integral $\textbf{r}(t) = \begin{pmatrix} 3t \\ 4 \end{pmatrix}$ over $[1, 3]$.

    $\begin{pmatrix} 9 \\ 8 \end{pmatrix}$. Integrate $3t$ to get $\frac{3}{2}(3^2-1^2)=9$ and $4$ to get $4(3-1)=8$.

17. What is the integral of the zero vector function $\textbf{0}$ over any interval $[a, b]$?

    $\textbf{0}$. The zero vector integrated over any interval is the zero vector.

18. Which integral property allows you to integrate each component separately in a vector-valued function?

    Linearity of integration. Integration distributes over vector addition and scalar multiplication.

19. Evaluate the integral of $\textbf{r}(t) = \begin{pmatrix} \text{cosh}(t) \\ \text{sinh}(t) \end{pmatrix}$ over $[0, 1]$.

    $\begin{pmatrix} \text{sinh}(1) \\ \text{cosh}(1) - 1 \end{pmatrix}$. Integrate hyperbolic functions: $\cosh(t) \to \sinh(t)$ and $\sinh(t) \to \cosh(t)$.

20. What is the integral of constant vector-valued function $\textbf{c}$ over $[a, b]$?

    $(b-a)\textbf{c}$. A constant vector integrated over an interval equals the vector times the interval length.

21. How is the integral of a vector-valued function affected by scalar multiplication?

    Scalar multiplies the integral. The scalar factor can be pulled out of the integral.

22. Which theorem is used to evaluate the integral of a vector-valued function over an interval $[a, b]$?

    Fundamental Theorem of Calculus. Applies to vector functions by evaluating at bounds and subtracting.

23. How do you find the integral of $\textbf{r}(t) = \begin{pmatrix} t^2 \\ t^3 \end{pmatrix}$ over $[0, 1]$?

    $\begin{pmatrix} \textstyle\frac{1}{3} \\ \textstyle\frac{1}{4} \end{pmatrix}$. Apply power rule: $\int t^n dt = \frac{t^{n+1}}{n+1}$ to each component.

24. Find the integral of $\textbf{r}(t) = \begin{pmatrix} t^3 \\ t^4 \end{pmatrix}$ over $[0, 1]$.

    $\begin{pmatrix} \textstyle\frac{1}{4} \\ \textstyle\frac{1}{5} \end{pmatrix}$. Use power rule: $\int t^3 dt = \frac{t^4}{4}$ and $\int t^4 dt = \frac{t^5}{5}$.

25. What is the integral of a linear vector function $\textbf{r}(t) = \begin{pmatrix} at + b \\ ct + d \end{pmatrix}$?

    $\begin{pmatrix} \textstyle\frac{1}{2}at^2 + bt \\ \textstyle\frac{1}{2}ct^2 + dt \end{pmatrix} + \textbf{C}$. Apply power rule to each linear component separately.

26. Find the integral of $\textbf{r}(t) = \begin{pmatrix} t^3 \\ t^4 \end{pmatrix}$ over $[0, 1]$.

    $\begin{pmatrix} \textstyle\frac{1}{4} \\ \textstyle\frac{1}{5} \end{pmatrix}$. Use power rule: $\int t^3 dt = \frac{t^4}{4}$ and $\int t^4 dt = \frac{t^5}{5}$.

27. What is the indefinite integral of $\textbf{r}(t) = \begin{pmatrix} 1 \\ t \end{pmatrix}$?

    $\begin{pmatrix} t \\ \textstyle\frac{1}{2}t^2 \end{pmatrix} + \textbf{C}$. Antiderivatives of $1$ and $t$ with constant vector added.

28. Express the integral $\textbf{r}(t) = \begin{pmatrix} t^2 \\ \sin(t) \end{pmatrix}$ over $[0, 2]$ in component form.

    $\begin{pmatrix} \textstyle\frac{8}{3} \\ 1 - \cos(2) \end{pmatrix}$. Integrate $t^2$ to get $\frac{8}{3}$ and $\sin(t)$ to get $1 - \cos(2)$.

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