For which of the following data sets would you expect a histogram to be skewed to the right? For which would it be skewed to the left?

$\textbf{a.}\hspace{10pt}$ The lengths of the words in a book

$\textbf{b.}\hspace{10pt}$ Dates of coins in circulation

$\textbf{c.}\hspace{10pt}$ Scores of students on an easy exam

Consider the chilled water circulation system of earlier problem. Select pumps that may be combined in parallel to supply the total flow requirement. Calculate the power required for $3$ pumps in parallel. Also calculate the volume flow rates and power required when only $1$ or $2$ of these pumps operates.

Consider the two-dimensional flow u = -Ax, υ = Ay, where A is a constant. Evaluate the circulation Γ around the rectangular closed curve defined by (x, y) = (1, 1), (4, 1), (4, 3), and (1, 3). Interpret your result, especially vis-ã -vis the velocity potential.

For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F=

$$\left\langle x + y ^ { 2 } , x ^ { 2 } - y \right\rangle$$

, where R=

$$\left\{ ( x , y ) : 3 y ^ { 2 } \leq x \leq 36 - y ^ { 2 } \right\}$$

For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F=$\langle - y , x \rangle$; R is the annulus $\{ ( r , \theta ) : 1 \leq r \leq 3,0 \leq \theta \leq 2 \pi \}$.

Given the vector field $\mathbf{G}=(16 x y-z) \mathbf{a}_{x}+8 x^{2} \mathbf{a}_{y}-x \mathbf{a}_{z}$ (a) Is G irrotational (or conservative)? (b) Find the net flux of G over the cube $0<x, y, z<1$. (c) Determine the circulation of G around the edge of the square z = 0, 0 < x, y < 1. Assume anticlockwise direction.

Use Green's Theorem to evaluate the following line integrals. Assume all curves are oriented counterclockwise. A sketch is helpful. The circulation line integral of $\mathbf{F}=\left\langle x^{2}+y^{2}, 4 x+y^{3}\right\rangle$, where C is the boundary of $\{(x, y): 0 \leq y \leq \sin x, 0 \leq x \leq \pi\}$

Circle the BEST answer.
Which statement about positioning is true?
a Re-positioning prevents pressure ulcers and contractures.
b Circulation is not affected by positioning.
c Position changes are avoided if moving causes pain.
d Persons in chairs are re-positioned less often than those in bed.

Use Green's theorem to find the counterclockwise circulation and outward flux for the field $\mathbf{F}$ and curve $C$. \

$$\mathbf{F}=\left(y^2-x^2\right) \mathbf{i}+\left(x^2+y^2\right) \mathbf{j}$$

$C$ : The triangle bounded by $y=0, x=3$, and $y=x$

The upper half of the ellipsoid $\frac{1}{2} x^2+\frac{1}{2} y^2+z^2=1$ intersects the cylinder $x^2+y^2-y=0$ in a curve $C$. Calculate the circulation of $\mathbf{v}=y^3 \mathbf{i}+\left(x y+3 x y^2\right) \mathbf{j}+z^4 \mathbf{k}$ around $C$ by using Stokes's theorem.

For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F=$\nabla \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$, where R is the half annulus $\{ ( r , \theta ) : 1 \leq r \leq 3,0 \leq \theta \leq \pi \}$

A smooth vector field $\vec{G}$ has curl $\vec{G} (0, 0, 0) = 2\vec{i} − 3\vec{j} + 5\vec{k}$. Estimate the circulation around a circle of radius $0.01$ centered at the origin in each of the following planes:

$xy$-plane, oriented counterclockwise when viewed from the positive $z$-axis.

Use Green's theorem to find the counterclockwise circulation and outward flux for the field $\mathbf{F}$ and curve $C$. \

$$\mathbf{F}=\left(x^2+4 y\right) \mathbf{i}+\left(x+y^2\right) \mathbf{j}$$

$C$ : The square bounded by $x=0, x=1, y=0, y=1$

Use Green’s Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. The circulation line integral of F=

$$\left\langle x ^ { 2 } + y ^ { 2 } , 4 x + y ^ { 3 } \right\rangle$$

, where C is the boundary of

$$\{ ( x , y ) : 0 \leq y \leq \sin x , 0 \leq x \leq \pi \}$$