Answer
True
Work Step by Step
The integral of a vector-valued function \(\mathbf{r}(t)\) is defined to be a vector whose components are the integral of each component of \(\mathbf{r}(t)\). So if
\(\mathbf{r}(t) = x(t)i + y(t)j + z(t)k\),
then
\[ \int_{a}^{b} \mathbf{r}(t) \, dt = \left( \int_{a}^{b} x(t) \, dt \right)i + \left( \int_{a}^{b} y(t) \, dt \right)j + \left( \int_{a}^{b} z(t) \, dt \right)k \] In the case that \(\mathbf{r}(t)\) is in 2-space, it is the same but without the \(k\) component.
Result: True statement
