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Problem 44

In Exercises $43-46,$ use a CAS to perform the fo…

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Problem 43

In Exercises $43-46,$ use a CAS to perform the following steps to evaluate the line integrals.
a. Find $d s=|\mathbf{v}(t)| d t$ for the path $\mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+$ $k(t) \mathbf{k} .$
b. Express the integrand $f(g(t), h(t), k(t))|\mathbf{v}(t)|$ as a function of the parameter $t .$
c. Evaluate $\int_{C} f d s$ using Equation $(2)$ in the text.
$$f(x, y, z)=\sqrt{1+30 x^{2}+10 y} ; \quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+3 t^{2} \mathbf{k} 0 \leq t \leq 2$$

Answer

a) $d s=\left(\sqrt{1+40 t^{2}}\right) d t$
b) $f(g(t), h(t), k(t))|v(t)|=\left(\sqrt{1+40 t^{2}}\right)\left(\sqrt{1+40 t^{2}}\right)$
c) $\int_{C} f \cdot d s \approx 108.67$



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