8.3.2Can I build a model?

Cross-Sections Lab: Functions Given

8-109.

Your team will construct a model of a solid with a known cross-section. You will have about $30$ minutes to complete this task. It is important to divide the work equally so that you will finish in time. Follow the steps below.

Build a Cross-Sections Model

Directions: Choose a base set of functions and a cross-section shape. Once a base and a shape are taken, no other team may choose the same combination. Tell your teacher your choice, such as 5D.

1. Base: $y =\sqrt { 9 - x ^ { 2 } } ; y = 0$
Cross-section perpendicular to the $x$ -axis.

A. Square

2. Base: $y =\sqrt { 9 - x ^ { 2 } } ; y = 0$
Cross-section perpendicular to the $y$ -axis.

B. Rectangle where the height is half the base

3. Base: $y = 2 | x | - 3 ; y = 3$
Cross-section perpendicular to the $x$ -axis.

C. Rectangle where the height is twice the base

4. Base: $y = x$ ; $y = -3$ ; $x = 3$
Cross-section perpendicular to the $x$ -axis.

D. Semi-circle

5. Base: $y = \sqrt { x + 1 } ; y = 0 ; x = 3$
Cross-section perpendicular to the $x$ -axis.

E. Isosceles right triangle where the triangle base is a leg

6. Base: $y = 3 \sin x$ ; $x = 3$ ; $x = -3$
Cross-section perpendicular to the $x$ -axis.

F. Equilateral triangle

7. Base: $y = \sqrt { x + 1 } ; y = 0 ; x = 3$
Cross-section perpendicular to the $y$ -axis.

G. Right trapezoid where the height and top base are half the bottom base

8. Base: $y = x$ ; $y = -3$ ; $x = 3$
Cross-section perpendicular to the $y$ -axis

H. Quarter circle

9. Base: $y = 2 | x | - 3 ; y = 3$
Cross-section perpendicular to the $y$ -axis.

I. Sector ( $\frac { 1 } { 8 }$ of a circle)

Obtain your materials: Lesson 8.3.2 Resource Page, cardstock, tape, ruler, and scissors.
Draw the base on the resource page.
Decide how the volume of each cross-section will be calculated.
Each team member takes either an $x$ - or a $y$ - value depending on the orientation of the solid and cuts out a cross-section of the shape that “fits” the base at that particular value. You may want to add a tab at the base, split it in half, folding the tabs in opposite direction so that the shape is upright when you tape.
When at least eight cross-sections are cut, two team members assemble the solid.
The remaining members use graphing calculators to compute the volume.
Be sure to include the integral as well as the answer on your model.

8-110.

Given that a typical cross-section of a solid taken perpendicular to the $x$ -axis has an area of $A(x) = \ln(x)$ , set up an integral to calculate the volume of this solid from $[a, b]$ . Does it matter what the shape of the solid is? Explain. Homework Help ✎

8-111.

Uh-oh! Arlen needs your help. He is trying to write an integral for the volume of square cross-sections perpendicular to the $y$ -axis where the base is the region bounded by the $y$ -axis and $y = x^2$ over the interval $0 ≤ x ≤ 3$ . Help him set up this integral. Homework Help ✎

8-112.

The base of a solid with square cross-sections perpendicular to the $x$ -axis is bounded by $f(x) =\sqrt { x }$ and the $x$ -axis. Sketch the solid formed and set up an integral to calculate the volume for $0 ≤ x ≤ 4$ . Homework Help ✎

8-113.

The base of a solid with square cross-sections perpendicular to the $x$ -axis is bounded by the curves $y = -x^2 + 4$ and $y = x^2 - 4$ . Set up an integral to calculate the volume. Do not solve. 6-113 HW eTool. Homework Help ✎

8-114.

The length of a rectangle is increasing at a rate of $6$ cm/sec and its width is increasing at a rate of $2$ cm/sec. When the length is $30$ cm and the width is $15$ cm, how fast is the area of the rectangle increasing? Homework Help ✎

8-115.

Evaluate the following integrals without using your calculator. Be sure to change the bounds if you use $u$ -substitution. Homework Help ✎

Compute without a calculator

$\int _ { 1 } ^ { 5 } \frac { x } { \sqrt { x ^ { 2 } - 1 } } d x$

$\int _ { 0 } ^ { 1 } x \sqrt { 1 - x ^ { 2 } } d x$

$\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 1 } d x$

$\int _ { 0 } ^ { 2 } 3 x e ^ { - x ^ { 2 } } d x$

8-116.

Paolo wants to surprise his girlfriend Jessica (who loves chocolate) by baking and decorating a cake for her to celebrate their anniversary. The frosting bag is in the shape of a right cone with a radius equal to its height. He squeezes the bag with constant pressure so that he applies $15 \text{ cm}^3$ of frosting per minute. Homework Help ✎

How much frosting will Paolo have put on the cake after $42$ seconds?
How fast is the radius of the frosting bag changing at the moment the radius is $1.7$ cm?
Will Jessica like the cake?

8-117.

Suppose $f(g(x)) = x$ , such that $f$ and $g$ are differentiable. If $f(1) = 3$ and $f ^\prime(1) = 2$ , determine the value of $g^\prime(3)$ . Homework Help ✎

8-118.

A particle is moving along the $y$ -axis with a position function $y(t) =\frac { 1 } { 2 }t^2 + 2t + 1$ . Determine all times when the particle is traveling at its average rate between $0$ and $8$ seconds. Homework Help ✎

8-119.

Evaluate the following limits. Homework Help ✎

$\lim\limits_ { x \rightarrow 6 } \frac { \operatorname { ln } ( x - 4 ) - \operatorname { ln } ( 2 ) } { x - 6 }$

$\lim\limits _ { x \rightarrow 25 } \frac { \sqrt { 2 x - 1 } - 7 } { x - 25 }$

$\lim\limits _ { x \rightarrow \infty } \frac { \operatorname { sin } ^ { 2 } ( x ) } { x }$

$\lim\limits _ { x \rightarrow \infty } \frac { 3 ^ { x } + 7 x } { e ^ { x } + 10 x }$

Calculus Third Edition

Build a Cross-Sections Model