8.3.1Can I build a model?

Cross-Sections Lab: General Case

8-97.

Are there other kinds of solids that are not revolved about a particular line where we can slice and accumulate the volumes? Will any shape work? Do the slices need to be circular? What conditions make a solid a good candidate for slicing and accumulating the volumes of each slice? With your team, discuss each of the solids below. For which solids do you believe accumulation of slices is a good method for computing the volume? Why?

8-98.

The Goodslice Baking Company is famous for making sandwich bread. When sliced, each piece of bread forms a square. Although each slice has the same thickness, the area of the square is not the same for all slices.

Draw an example of a slice of Goodslice’s sandwich bread on your paper.
Assume that there are $n$ slices. If $A(i)$ determines the area of the $i^{th}$ slice, use summation notation to represent the volume of the entire loaf of bread. Let the thickness of each slice be $∆x$ .
With your team, discuss how to compute the exact volume of the loaf. Write down your conclusions.

8-99.

Your team will be building a three-dimensional model of a solid with similar cross-sections.

On a piece of cardstock, draw a two-dimensional shape roughly the size of a sticky note folded in half. The more irregular your shape is, the more interesting your cross-sections will be.
Your teacher will provide your team with a cross-section description. Using play dough, build on top of your two-dimensional base shape so that every cross-section fits the cross-section description given to your team.
Be prepared to present your model to the class. During your presentation be sure to:
1. Describe the shape of your model’s cross-section.
2. Explain all of the ways you can slice to get the cross-section shape. Discuss slicing directions that will not give the cross-section shape.
3. Demonstrate your shape by physically removing a “typical” cross-section.

8-100.

Summarize a method for calculating the volume of a “typical” cross-section. How can you approximate the total volume of your solid?

8-101.

Cavalieri’s principle states that if two solids have equal height and all corresponding cross-sections have the same area, then the solids have the same volume. This relationship holds even when the two solids do not have the same shape. Examine a deck of cards to help explain why this must be true. Homework Help ✎

8-102.

The graph at right shows $f(x)=\frac { 1 } { 2 }x \cos(x) + 4$ for $0 ≤ x ≤ 5$ . Write an integral that will compute the volume of the solid when this region is rotated about: Homework Help ✎

The $x$ -axis.
The $y$ -axis.
The line $x = 6$ .
The line $y = -7$ .

8-103.

Multiple Choice: $\frac { d } { d x } [ \operatorname { arccos } ( 3 x ) ] =$ Homework Help ✎

$\frac { 3 } { \sqrt { 1 - x ^ { 2 } } }$

$\frac { - 1 } { 3 \operatorname { sin } ( 3 x ) }$

$\frac { 3 } { \sqrt { 1 - 3 x ^ { 2 } } }$

$\frac{3}{\sqrt{1-9x^2}}$

$\frac{-3}{\sqrt{9x^2-1}}$

8-104.

Multiple Choice: A particle has a velocity of $v(t) = t + \sin(t)$ over the interval $0 ≤ t ≤ π$ . The average velocity of the particle is: Homework Help ✎

$\frac { \pi } { 2 } + \frac { 2 } { \pi }$

$\frac { \pi ^ { 2 } } { 2 }$

$\frac { 1 } { 2 }$

$\frac { 2 } { \pi }$

8-105.

Multiple Choice: If $F ( x ) = \int _ { 0 } ^ { x } \sqrt { 9 - t } d t$ , which of the following statements are true? Homework Help ✎

$F^\prime(5) = 2$

$F(−7) > F(5)$

$F$ is concave downward

I only

II only

I and III

II and III

I and II

8-106.

Multiple Choice: The population $P$ of a city is growing according to the equation $\frac { d P } { d t }= 0.02P + 357$ . The current population of the city is $18,000$ . Assuming the same pattern of growth, what will be the population be in six years? Homework Help ✎

$18,351$

$20,250$

$21,348$

$22,571$

$23,374$

8-107.

Multiple Choice: If $y = 3x − 12$ , then the minimum product for $xy$ is: Homework Help ✎

$-9$

$-12$

$-18$

8-108.

Multiple Choice: Given the graph of $y = f (t)$ at right and $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$ . Which of the graphs below could be the graph of $y = g(x)$ ? Homework Help ✎

X axis with points labeled, A & b, Curve labeled, f of x, starting on the positive y axis, concave down, changing to concave up as it passes through the x axis at point labeled, a, turning up half way between a & b, & stopping at point on x axis labeled, b.

Calculus Third Edition