📚 Topic Summary
Integrating vector-valued functions involves integrating each component function separately. If $\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle$, then $\int \mathbf{r}(t) \, dt = \langle \int f(t) \, dt, \int g(t) \, dt, \int h(t) \, dt \rangle + \mathbf{C}$, where $\mathbf{C}$ is the constant vector of integration. Definite integrals are computed similarly, evaluating the integral of each component function over the given interval.
📝 Part A: Vocabulary
Match the following terms with their definitions:
| Term |
Definition |
| 1. Vector-Valued Function |
A. A vector that represents the constant of integration. |
| 2. Component Function |
B. The process of finding the antiderivative of a function. |
| 3. Constant Vector of Integration |
C. A function that returns a vector as its output. |
| 4. Integration |
D. A function that makes up one part of a vector-valued function. |
| 5. Antiderivative |
E. A function whose derivative is the original function. |
✍️ Part B: Fill in the Blanks
When integrating a vector-valued function, you integrate each ________ function separately. The result is another vector-valued function plus a ________ vector of ________. This constant vector is similar to the constant 'C' in single-variable calculus but applies to each ________.
🤔 Part C: Critical Thinking
Explain why the constant of integration for a vector-valued function is a vector and not just a scalar. What does this imply about the family of antiderivatives for a vector-valued function?