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.

## Discussion

## Video Transcript

Oh, has function this to assign See to co sign t 40 from zero to well, your pie. So the march of the derivative would be too. Using the identity of the children is a function. We're the nine to go f oversee. Forget the data. We're gonna have zero to have high. I'm sorry. 0 to 1 0/4 Bi, You saw that? We have for sun t squared on this tickles nt Did she? Tom's too. So the answer is plying minus two minus to Times Square to

## Recommended Questions

In Exercises $27 - 30$ , integrate $f$ over the given curve.

$$

\begin{array} { l } { f ( x , y ) = x + y , \quad C : \quad x ^ { 2 } + y ^ { 2 } = 4 \text { in the first quadrant from } } \\ { ( 2,0 ) \text { to } ( 0,2 ) } \end{array}

$$

In Exercises $27 - 30 ,$ integrate $f$ over the given curve.

$$

\begin{array} { l } { f ( x , y ) = \left( x + y ^ { 2 } \right) / \sqrt { 1 + x ^ { 2 } } , \quad C : \quad y = x ^ { 2 } / 2 \text { from } ( 1,1 / 2 ) \text { to } } \\ { ( 0,0 ) } \end{array}

$$

In Exercises $27 - 30 ,$ integrate $f$ over the given curve.

$$f ( x , y ) = x ^ { 3 } / y , \quad C : \quad y = x ^ { 2 } / 2 , \quad 0 \leq x \leq 2$$

Integrate $f$ over the given curve.

$f(x, y)=x+y, \quad C: \quad x^{2}+y^{2}=4$ in the first quadrant from (2,0) to (0,2)

Integrate $f$ over the given curve.

$f(x, y)=x^{2}-y, \quad C: \quad x^{2}+y^{2}=4$ in the first quadrant from (0,2) to $(\sqrt{2}, \sqrt{2})$

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.

$$

\int_{0}^{2 \sqrt{\ln 3}} \int_{y / 2}^{\sqrt{\ln 3}} e^{x^{2}} d x d y

$$

Integrate $f$ over the given curve.

$f(x, y)=\left(x+y^{2}\right) / \sqrt{1+x^{2}}, \quad C: \quad y=x^{2} / 2$ from $(1,1 / 2)$ to (0,0)

In Exercises 25 and $26,$ integrate $f$ over the given region.

$$\begin{array}{l}{\text { Square } f(x, y)=1 /(x y) \quad \text { over } \quad \text { the square } \quad 1 \leq x \leq 2} \\ {1 \leq y \leq 2}\end{array}$$

In Exercises $29-34,$ calculate the derivative.

$$

\frac{d}{d x} \int_{x^{2}}^{x^{4}} \sqrt{t} d t

$$

Hint for Exercise $32 : F(x)=A\left(x^{4}\right)-A\left(x^{2}\right)$

Integrate $f$ over the given curve.

$f(x, y)=x^{3} / y, \quad C: \quad y=x^{2} / 2, \quad 0 \leq x \leq 2$