Section 13.2: Derivatives and Integrals of Vector Functions

The definite integral of a continuous vector function \(\mathbf{r}(t)\) can be defined in much the same way as for real-valued functions except that the integral is a vector. But then we can express the integral of \(\mathbf{r}\) in terms of the integrals of its component functions f, g, and h as follows.

\[ \int_a^b \mathbf{r}(t) dt = \lim_{n \to \infty} \sum_{i=1}^n \mathbf{r}(t_i^*) \Delta t \] \[ = \lim_{n \to \infty} \left[ \left( \sum_{i=1}^n f(t_i^*) \Delta t \right)\mathbf{i} + \left( \sum_{i=1}^n g(t_i^*) \Delta t \right)\mathbf{j} + \left( \sum_{i=1}^n h(t_i^*) \Delta t \right)\mathbf{k} \right] \] and so \[ \int_a^b \mathbf{r}(t) dt = \left( \int_a^b f(t) dt \right)\mathbf{i} + \left( \int_a^b g(t) dt \right)\mathbf{j} + \left( \int_a^b h(t) dt \right)\mathbf{k} \] This means that we can evaluate an integral of a vector function by integrating each component function. We can extend the Fundamental Theorem of Calculus to continuous vector functions as follows: \[ \int_a^b \mathbf{r}(t) dt = \mathbf{R}(t) \Big]_a^b = \mathbf{R}(b) - \mathbf{R}(a) \] where \(\mathbf{R}\) is an antiderivative of \(\mathbf{r}\), that is, \(\mathbf{R}'(t) = \mathbf{r}(t)\). We use the notation \(\int \mathbf{r}(t) dt\) for indefinite integrals (antiderivatives).

EXAMPLE 5 If \(\mathbf{r}(t) = 2\cos t \mathbf{i} + \sin t \mathbf{j} + 2t \mathbf{k}\), then \[ \int \mathbf{r}(t) dt = \left( \int 2\cos t dt \right)\mathbf{i} + \left( \int \sin t dt \right)\mathbf{j} + \left( \int 2t dt \right)\mathbf{k} \] \[ = 2\sin t \mathbf{i} - \cos t \mathbf{j} + t^2 \mathbf{k} + \mathbf{C} \] where \(\mathbf{C}\) is a vector constant of integration, and \[ \int_0^{\pi/2} \mathbf{r}(t) dt = [2\sin t \mathbf{i} - \cos t \mathbf{j} + t^2 \mathbf{k}]_0^{\pi/2} = 2\mathbf{i} + \mathbf{j} + \frac{\pi^2}{4}\mathbf{k} \]