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Chapter 12: Problem 6

How do you evaluate \(\int_{a}^{b} \mathbf{r}(t) d t ?\)

Short Answer

Expert verified
Question: Evaluate the definite integral of the vector function \(\mathbf{r}(t) = \langle{f(t), g(t), h(t)}\rangle\) from \(t=a\) to \(t=b\). Answer: To evaluate the definite integral of the vector function \(\mathbf{r}(t)\) from \(t=a\) to \(t=b\), integrate each component separately and combine the results to form a single vector: $$\int_{a}^{b} \mathbf{r}(t) dt = \langle{F(b) - F(a), G(b) - G(a), H(b) - H(a)}\rangle$$ where \(F(t)\), \(G(t)\), and \(H(t)\) are the antiderivatives of \(f(t)\), \(g(t)\), and \(h(t)\), respectively.

Step by step solution

01

Identify the components of the vector function

Write the vector function \(\mathbf{r}(t)\) as \(\mathbf{r}(t) = \langle{f(t), g(t), h(t)}\rangle\).
02

Integrate each component separately

Integrate the functions \(f(t)\), \(g(t)\), and \(h(t)\) with respect to \(t\) over the interval from \(a\) to \(b\): $$\int_{a}^{b} f(t) dt = F(b) - F(a)$$ $$\int_{a}^{b} g(t) dt = G(b) - G(a)$$ $$\int_{a}^{b} h(t) dt = H(b) - H(a)$$ where \(F(t)\), \(G(t)\), and \(H(t)\) are the antiderivatives of \(f(t)\), \(g(t)\), and \(h(t)\), respectively.
03

Write the result in vector form

Combine the results from the previous step into a single vector: $$\int_{a}^{b} \mathbf{r}(t) dt = \langle{F(b) - F(a), G(b) - G(a), H(b) - H(a)}\rangle$$

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