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.

$$f(x, y, z)=\left(1+\frac{9}{4} z^{1 / 3}\right)^{1 / 4} ; \quad \mathbf{r}(t)=(\cos 2 t) \mathbf{i}+(\sin 2 t) \mathbf{j}+ t^{5 / 2} \mathbf{k}, \quad 0 \leq t \leq 2 \pi$$

## Discussion

## Video Transcript

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## Recommended Questions

Center of mass and moments of inertia for wire with variable density Find the center of mass and the moments of inertia about the coordinate axes of a thin wire lying along the curve

$$

\mathbf { r } ( t ) = t \mathbf { i } + \frac { 2 \sqrt { 2 } } { 3 } t ^ { 3 / 2 } \mathbf { j } + \frac { t ^ { 2 } } { 2 } \mathbf { k } , \quad 0 \leq t \leq 2

$$

if the density is $\delta = 1 / ( t + 1 )$

Find the center of mass and the moments of inertia about the coordinate axes of a thin wire lying along the curve

$$\mathbf{r}(t)=t \mathbf{i}+\frac{2 \sqrt{2}}{3} t^{3 / 2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k}, \quad 0 \leq t \leq 2$$

if the density is $\delta=1 /(t+1)$.

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Finding a center of mass and moment of inertia Find the center of mass and moment of inertia about the $x$ -axis of a thin plate

bounded by the curves $x=y^{2}$ and $x=2 y-y^{2}$ if the density at

the point $(x, y)$ is $\delta(x, y)=y+1$

Center of mass, moment of inertia Find the center of mass

and the moment of inertia about the $y$ -axis of a thin plate bounded by the line $y=1$ and the parabola $y=x^{2}$ if the density is

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If a wire with linear density $ \rho(x, y) $ lies along a plane curve $ C $, its $ \textbf{moments of inertia} $ about the $ x $- and $ y $-axes are defined as

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Find the moments of inertia for the wire in Example 3.

Find the center of mass of a thin wire lying along the curve $\mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+$ $(2 / 3) t^{3 / 2} \mathbf{k}, 0 \leq t \leq 2,$ if the density is $\delta=3 \sqrt{5+t}$.

Center of mass of a curved wire A wire of density $\delta(x, y, z)=15 \sqrt{y+2}$ lies along the curve $\mathbf{r}(t)=\left(t^{2}-1\right) \mathbf{j}+$ $2 t \mathbf{k},-1 \leq t \leq 1 .$ Find its center of mass. Then sketch the curve and center of mass together.

Center of mass of a curved wire $A$ wire of density $\delta ( x , y , z ) = 15 \sqrt { y + 2 }$ lies along the curve $\mathbf { r } ( t ) = \left( t ^ { 2 } - 1 \right) \mathbf { j } +$ $2 t \mathbf { k } , - 1 \leq t \leq 1 .$ Find its center of mass. Then sketch the curve and center of mass together.

If a wire with linear density $ \rho(x, y, z) $ lies along a space curve $ C $, its $ \textbf{moments of inertia} $ about the $ x $-, $ y $-, and $ z $-axes are defined as

$$ I_x = \int_C (y^2 + z^2) \rho(x, y, z) ds $$

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$$ I_z = \int_C (x^2 + y^2) \rho(x, y, z) ds $$

Find the moments of inertia for the wire in Exercise 35.