Calculus III Advanced (Course) (15.03) (Homework)

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Section 15.3 Homework

From Calculus 10e by Larson and Edwards, p. 1072. Exercises 1, 16, 26, 36

Exercise 15.3.1 Evaluating a Line Integral for Different Parametrizations

Show that the value for \(\int_{C} \textbf{F} \cdot d\textbf{r}\) is the same for each parametric representation.

\( \textbf{F}(x,y)=x^{2}\textbf{i}+xy\textbf{j}\)
(a) \(\textbf{r}_{1}(t) = t\textbf{i}+t^{2}\textbf{j}\), \(0 \leqslant t \leqslant 1\)
(b) \(\textbf{r}_{2}(\theta) = \sin \theta\textbf{i}+\sin^{2} \theta \textbf{j}\), \(0 \leqslant \theta \leqslant \pi/2\)

Solution

(a) \(d\textbf{r} = 1+2t\) and \( \textbf{F}(x,y)=t^{2}\textbf{i}+ t^{3}\textbf{j}\)

\(\left(t^{2}+ t^{3}\right) \left(1+2t \right) = t^{2}+3t^{3}+2t^{4}\)
$$ \int_{0}^{1} t^{2}+3t^{3}+2t^{4} \:dt$$ $$ \left. \frac{1}{3} t^{3} + \frac{3}{4}t^{4}+ \frac{2}{5}t^{5} \right]_{0}^{1} = \frac{89}{60}$$

Exercise 15.3.16 Evaluating a Line Integral for a Vector Field

Exercise 15.3.26 Using the Fundamental Theorem of Line Integrals

Exercise 15.3.36 Work

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Parent Article: Calculus III Advanced (Course)