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: 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 2: 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 3: 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 4: 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 5: Evaluate the integral of $\textbf{r}(t) = \begin{pmatrix} \text{sinh}(t) \\ \text{cosh}(t) \end{pmatrix}$ over $[0, 1]$.

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

Flashcard 6: 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 7: 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 8: 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 9: State the formula for integrating a vector-valued function with respect to a parameter $u$.

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

Flashcard 10: 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 11: 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 12: 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 13: State the relationship between integration and differentiation for vector-valued functions.

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

Flashcard 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: 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 31: 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 32: 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 33: 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 34: 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 35: 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 36: 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 37: Evaluate the integral of $\textbf{r}(t) = \begin{pmatrix} \text{sinh}(t) \\ \text{cosh}(t) \end{pmatrix}$ over $[0, 1]$.

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

Flashcard 38: 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 39: State the relationship between integration and differentiation for vector-valued functions.

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

Flashcard 40: 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 41: 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 42: 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 43: 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 44: 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 45: State the formula for integrating a vector-valued function with respect to a parameter $u$.

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

Flashcard 46: 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 47: 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 48: 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.