Find the volume of a region

$$\mathcal { W }$$

if

$$\iint _ { \partial \mathcal { W } } \left\langle x + x y + z , x + 3 y - \frac { 1 } { 2 } y ^ { 2 } , 4 z \right\rangle \cdot d \mathbf { S } = 16$$

Find the area of the region. Use a graphing utility to verify your result.

$$\int_{\pi / 2}^{2 \pi / 3} \sec ^2\left(\frac{x}{2}\right) d x$$

Solve the following problem and choose correct answer from the given options.

$$\int x \sqrt{16-3 x^2} d x=$$

(A) $\frac{2}{3}\left(16-3 x^2\right)^{3 / 2}+C$

(B) $-\frac{1}{4}\left(16-3 x^2\right)^{3 / 2}+C$

(C) $-\frac{1}{9}\left(16-3 x^2\right)^{3 / 2}+C$

(D) $\frac{1}{9}\left(16-3 x^2\right)^{3 / 2}+C$

Which value best approximates the area of the region bounded by the graph of

$$f(x)=\sin \frac{\pi x}{4}$$

and the $x$-axis over the interval $[0,4]$ ?

A. 2

B. $2.5$

C. 3

D. $\pi$

Convert the temperature to degrees Fahrenheit. 60°C

(a) Use cylindrical shells to find the volume of the solid that is generated when the region under the curve

$$y=x^3-3 x^2+2 x$$

over $[0,1]$ is revolved about the $y$-axis. (b) For this problem, is the method of cylindrical shells easier or harder than the method of slicing discussed in the last section? Explain.

Solve each problem. See Example mentioned.

The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix A.

$$\begin{array}{llll} \text{Rolls (doz)}\\ \text{Bread (loaf) } \\ \text{Cake} \\ \text{Pie (crust) } \end{array} \overset{\text{Eggs} \quad \text{Flour} \quad \text{Sugar} \quad \text{Shortening} \quad \text{Milk}}{\left[\begin{array}{llllllll} 1 \qquad 4 \qquad \frac{1}{4} \qquad \frac{1}{4} \qquad 1 \\ 0 \qquad 3 \qquad 0 \qquad \frac{1}{4} \qquad 0 \\ 4 \qquad 3 \qquad 2 \qquad 1 \qquad 1 \\ 0 \qquad 1 \qquad 0 \qquad \frac{1}{3} \qquad 0 \end{array}\right]}=A$$

The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix B.

$$\begin{array}{llll} \text{Eggs}\\ \text{Flour} \\ \text{Sugar} \\ \text{Shortening} \\ \text{Milk}\end{array} \overset{\text{Cost}}{\overset{\text{Large Lot} \quad \text{Small Lot}}{\left[\begin{array}{rr} 5 & 5 \\ 8 & 10 \\ 10 & 12 \\ 12 & 15 \\ 5 & 6 \end{array}\right]}}=B$$

(a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options.

(b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1 \times 4$ matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders.

(c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Wareham High School, in Massachusetts, has about 900 students in grades 9-12. Read the information about a survey to assess student opinion about mandatory school uniforms.

a. Referring to the questions in the earlier problem, answer the questions that can be answered based on the information in the abstract above.

b. Does there appear to be any reason to find this survey untrustworthy? Explain.

c. Does the Assistant Principal have statistically significant evidence that a majority of students would have expressed a willingness to explore the possibility of uniforms if she had asked all of them? Select the best choice in each pair of options below.

Assuming that the proportion of all students who would have expressed this willingness is exactly/more than $50 \%$, we compute 1 - binomcdf(184,0.50,97) $\approx$ $0.2087$. Because this $P$-value is more/less than $5 \%$, the Assistant Principal does/does not have statistically significant evidence that the proportion of all students who are willing is exactly/greater than $50 \%$. In other words, if the true proportion is $50 \%$, it would/would not be very likely to get $\underline{50 / 53 \%}$ or even more who are willing in a sample of size $\underline{98 / 184}$.

Test for symmetry with respect to the polar axis.

$$r = 2 \cos 2 \theta$$

Using classes, design an online address book to keep track of the names, addresses, phone numbers, and dates of birth of family members, close friends, and certain business associates. Your program should be able to handle a maximum of 500 entries.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the curves $y=1 /\left(1+x^2\right), y=0, x=0, x=3 ;$ about the $y$-axis.

Refer to a dictionary or other resources if needed in order to complete the table .

$$\small{ \begin{array}{|l|l|l|l|} \hline \text{WORD}& \text{MENTOR SENTENCES}&\text{PREDICT MEANING}& \text{RELATED WORDS}\\ \hline \text{analyze} & 1. \text{The investigators will analyze} & &\text{analysis; analytical}\\ & \text{the scene for signs of arson.}& &\\ \text{ROOT:} & 2. \text{To anabyze a poem, start by}&&\\ \text{-lys- }& \text{examining its words and} &&\\ \text{"break down"} &\text{phrasing.}&&\\ \hline \text{subordinate} & 1. \text{It's important to show how}&&\\ & \text{subordinate ideas relate to the}\\ \text{ROOT:} & \text{main idea.}&&\\ \text{-ord-} & 2. \text{In her first job, she was in a}&&\\ \text{"order"} & \text{subordinate role, but she later}\\ & \text{became chief executive of the company.}\\ \hline \text{literal} & 1. \text{The original, literal meaning}&&\\ &\text{of "awful" is "full of awe," but}&&\\ \text{ROOT:} &\text{now it means "terrible.}"&&\\ \text{- liter-} & 2. \text{In her essay, she explains}&&\\ \text{"letter"}&\text{both the literal and symbolic}&&\\ &\text{meanings of the movie.}\\ \hline \text{determine} & 1. \text{The choices you make now}&&\\ &\text{could determine your future}&&\\ \text{ROOT:}&\text{options.}&&\\ \text{-term-}& 2. \text{We must do more than treat the}&&\\ \text{"end"} & \text{symptoms; we must determine}&&\\ &\text{the cause of the illness.}&&\\ \text{trivialize}& 1. \text{Politicians tend to trivialize}&&\\ &\text{issues that they do not consider}&&\\ \text{ROOT:}&\text{important.}&&\\ \text{- via -}&2. \text{Aeher laughed at Maya's error,}&&\\ \text{"way"; "path"}&\text{but he didn't mean to trivialize}&&\\ &\text{her struggle.}&&\\ \hline \end{array}}$$

Fill in details about movement of the reform movements, key leader, efforts made on behalf of the movement listed in abolition.

Evaluate the double integral $\iint _ { R } f ( x , y ) d A$ over the polar rectangular region D. $f ( x , y ) = \arctan \left( \frac { y } { x } \right)$, where $D = \{ ( r , \theta ) | 2 \leq r \leq 3 , \frac { \pi } { 4 } \leq \theta \leq \frac { \pi } { 3 } \}$