AP Calculus BC Flashcards: Integrating Vector Valued Functions

What this deck covers

This deck focuses on Integrating Vector Valued Functions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What rule applies when integrating vector functions with respect to a parameter?

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

Flashcard 2: What is the integral of a linear vector function $\textbf{r}(t) = \begin{pmatrix} at + b \ ct + d \end{pmatrix}$?

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

Flashcard 3: What is the integral of the zero vector function $\textbf{0}$ over any interval $[a, b]$?

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

Flashcard 4: How do you find the integral of $\textbf{r}(t) = \begin{pmatrix} t^2 \ t^3 \end{pmatrix}$ over $[0, 1]$?

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

Flashcard 5: What is the integral of constant vector-valued function $\textbf{c}$ over $[a, b]$?

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

Flashcard 6: Express the integral $\mathbf{r}(t) = \begin{pmatrix} t^2 \\ \sin(t) \end{pmatrix}$ over $[0, 2]$ in component form.

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

Flashcard 7: Evaluate the integral of $\textbf{r}(t) = \begin{pmatrix} \text{cosh}(t) \\ \text{sinh}(t) \end{pmatrix}$ over $[0, 1]$.

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

Flashcard 8: Determine the integral of $\textbf{r}(t) = \begin{pmatrix} t \ e^t \end{pmatrix}$ over $[0, 1]$.

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

Flashcard 9: How do you represent the antiderivative of a vector function $\textbf{r}(t)$?

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

Flashcard 10: What rule applies for integrating vector-valued functions with piecewise components?

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

Flashcard 11: Which theorem is used to evaluate the integral of a vector-valued function over an interval $[a, b]$?

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

Flashcard 12: Which integral property allows you to integrate each component separately in a vector-valued function?

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

Flashcard 13: How is the integral of a vector-valued function affected by scalar multiplication?

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

Flashcard 14: Find the integral of $\textbf{r}(t) = \begin{pmatrix} t^3 \ t^4 \end{pmatrix}$ over $[0, 1]$.

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

Flashcard 15: How does the integral of a vector-valued function change with a change of variable?

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

Flashcard 16: State the relationship between definite and indefinite integrals for vector functions.

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

Flashcard 17: What is the derivative of the integral of a vector-valued function $\textbf{r}(t)$?

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

Flashcard 18: In the context of vector-valued functions, what does $\textbf{C}$ represent?

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

Flashcard 19: Calculate the integral $\textbf{r}(t) = \begin{pmatrix} 3t \ 4 \end{pmatrix}$ over $[1, 3]$.

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

Flashcard 20: Evaluate the integral $\textbf{r}(t) = \begin{pmatrix} t \ t^2 \end{pmatrix}$ over $[1, 2]$.

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

Flashcard 21: Calculate the integral $\textbf{r}(t) = \begin{pmatrix} 3t \ 4 \end{pmatrix}$ over $[1, 3]$.

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

Flashcard 22: What is the indefinite integral of $\textbf{r}(t) = \begin{pmatrix} 1 \ t \end{pmatrix}$?

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

Flashcard 23: Evaluate the integral $\textbf{r}(t) = \begin{pmatrix} t \ t^2 \end{pmatrix}$ over $[1, 2]$.

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

Flashcard 24: How do you represent the antiderivative of a vector function $\textbf{r}(t)$?

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

Flashcard 25: Express the integral $\textbf{r}(t) = \begin{pmatrix} t^2 \\ \sin(t) \end{pmatrix}$ over $[0, 2]$ in component form.

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

Flashcard 26: What rule applies for integrating vector-valued functions with piecewise components?

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

Flashcard 27: What rule applies when integrating vector functions with respect to a parameter?

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

Flashcard 28: What is the integral of constant vector-valued function $\textbf{c}$ over $[a, b]$?

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

Flashcard 29: How do you find the integral of $\textbf{r}(t) = \begin{pmatrix} t^2 \ t^3 \end{pmatrix}$ over $[0, 1]$?

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

Flashcard 30: What is the derivative of the integral of a vector-valued function $\textbf{r}(t)$?

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