91Ó°ÊÓ

Question: Evaluate the integral \(\int_{a}^{b} \mathbf{r}(t) d t\) for the given vector-valued function \(\mathbf{r}(t)\). Answer: To evaluate the integral, follow these steps: 1. Write the vector-valued function in terms of its component functions, as \(\langle f(t), g(t), h(t) \rangle\). 2. Integrate each of the component functions over the interval \([a, b]\): \(\int_{a}^{b} f(t) dt\), \(\int_{a}^{b} g(t) dt\), and \(\int_{a}^{b} h(t) dt\). 3. Reassemble the integrated component functions into the final vector: \(\langle F, G, H \rangle\), where \(F\), \(G\), and \(H\) are the definite integrals of the component functions \(f(t)\), \(g(t)\), and \(h(t)\) over the interval \([a, b]\). Thus, the evaluated integral is \(\int_{a}^{b} \mathbf{r}(t) d t = \langle F, G, H \rangle\).