Calculus III Advanced (Course) (15.02) (Homework)

Section 15.2 Homework

From Calculus 10e by Larson and Edwards, p. 1061. Exercises 6, 10, 20, 22, 30, 38, 54.

Exercise 15.2.6 Finding a Piecewise Smooth Parametrization

Figure 1

Find a piecewise smooth parametrization for path \(C\), as shown in Figure 1. Solution $$ \textbf{r}(t) = \frac{1}{4}t\textbf{i}+\frac{1}{3}t\textbf{j}$$

Exercise 15.2.10 Evaluating a Line Integral

Evaluate the line integral along the given path.

$$\int_{C} 2xyz \: ds $$

\(C: \: \textbf{r}(t) = 12t\textbf{i}+5t\textbf{j}+84t\textbf{k}, \: 0 \leqslant t \leqslant 1\)

Solution Start with a smooth parametrization.

\(\textbf{r}^{\prime}(t) \)	\(=12+5+84 \)
	\(= \sqrt{12^{2}+5^{2}+84^{2} } =85\)

$$\int_{C} 2xyz \: ds $$	$$=\int_{0}^{1} (2)(12t)(5t)(84t)(85)\:dt $$
	$$= 856,800\int_{0}^{1} t^{3} \: dt$$
	$$= 856,800 \left[ \frac{1}{4} t^{4} \right]_{0}^{1}=214,200 $$

Exercise 15.2.20 Finding a Parametrization and Evaluating a Line Integral

Figure 2

(a) find a piecewise smooth parametrization for the path \(C\) shown in Figure 2. (b) evaluate $$\int_{C} (2x+y^{2}-z) \: ds $$ along \(C\). Solution \(C_{1}: \) \(x(t)=0 \) \(y(t)=t \) \(z(t)=0 \) \( 0 \leqslant t \leqslant 1 \) \(C_{2}: \) \(x(t)=0 \) \(y(t)=t-1 \) \(z(t)=t-1 \) \( 1 \leqslant t \leqslant 2 \) \(C_{3}: \) \(x(t)=0 \) \(y(t)=t-3 \) \(z(t)=t-3 \) \( 2 \leqslant t \leqslant 3 \) $$ \textbf{r}(t) = \left\{\begin{matrix} t\textbf{j} & 0 \leqslant t \leqslant 1 \\ (t-1)\textbf{j} + (t-1)\textbf{k} & 1 \leqslant t \leqslant 2 \\ (t-3)\textbf{j}+(t-3)\textbf{k} & 2 \leqslant t \leqslant 3\end{matrix}\right.$$

Exercise 15.2.22 Mass

Find the total mass for a spring with two turns and density \(\rho\) shaped as a circular helix

\(\textbf{r}(t) = 2 \cos t\textbf{i}+ 2 \sin t\textbf{j}+ t\textbf{k}, \: 0 \leqslant t \leqslant 4 \pi\).

\(\rho(x,y,z) = z.\)

Exercise 15.2.30 Evaluating a Line Integral of a Vector Field

Evaluate

$$ \int_{C} \textbf{F} \cdot \: d\textbf{r}$$

where \(C\) is represented by \(\textbf{r}(t)\).

\(\textbf{F}(x,y) = 3x\textbf{i}+4y\textbf{j}\)

\(C\): \(\textbf{r}(t)= t\textbf{i} + \sqrt{4-t^{2}}\textbf{j}, \: -2 \leqslant t \leqslant 2\).

Exercise 15.2.38 Work

Figure 3

Find the work done by the force field \(\textbf{F}\) on a particle moving along the path shown in Figure 3. \(\textbf{F}(x,y) = -y\textbf{i}-x\textbf{j}\) \(C\): counterclockwise along the semicircle \(y=\sqrt{4-x^{2}}\) from \((2,0)\) to \((-2,0)\).

Exercise 15.2.54 Evaluating a Line Integral in Differential Form

Evaluate the integral

$$\int_{C} 93y-x) \: dx + y^{2} \: dy $$

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Parent Article: Calculus III Advanced (Course)