8.2.2Which method is best?

Comparing the Disk and Shell Methods

8-76.

The region enclosed by $y = -(x - 3)^2 + 4$ , the $x$ -axis, and the line $x = 4$ is shown below.

Set up and evaluate an integral to calculate the area of the region.
Imagine rotating the region about the $y$ -axis. Use cylindrical shells to calculate the volume of the resulting solid.

Continuous piecewise, left downward parabola, vertex at the point (3, comma 4), starting @ (1, comma 0), ending at (4, comma 3), right vertical segment from (4, comma 3) to (4, comma 0), shaded region below pieces & above x axis.

8-77.

The graph at right shows the region in the first quadrant bounded by $f(x) = -x^2 + 2x + 3$ . Compare the different methods available to calculate the volume of the solid created when this region is revolved about the line $x = -1$ .

If washers are used, what happens to a typical slice over the interval $3 ≤ y ≤ 4$ ?
Why will shells be easier to use in this case?
What are the bounds of integration?
With your team, discuss what you know about the dimensions of a typical shell. What is the radius of a typical shell? The height?
Set up and evaluate the integral that will calculate the volume of the solid.

Downward parabola, vertex at (1, comma 4), starting at (0, comma 3), ending at (3, comma 0), shaded region below parabola, right of y axis, & above x axis, & vertical line at, x = negative 1.

8-78.

The shaded region below is bounded by $y = -x^2 + 2x + 3$ and $y = x + 1$ . Using cylindrical shells, set up the integral to calculate the volume of the solid created when the region is rotated about the line $x = -1$ .

Downward parabola, vertex at (1, comma 4), passing through the points (negative 1, comma 0) & (3, comma 0), increasing line intersecting the parabola at the points (negative 1, comma 0), & (2, comma 3), shaded region between the intersecting points, curve & line, & dashed vertical line at, x = negative 1.

8-79.

Investigators in Rosewell have come across a strange document containing design specifications for an alien craft. After years of translating the document, the research team has determined that it is the design of a transport vessel capable of carrying $120$ aliens. The ship can be formed by rotating the curve $y = 5x^{0.5} - 0.1x^{1.5}$ about the $x$ -axis, where $x$ and $y$ are in feet. Further evidence suggests that the alien life forms represents $10\%$ of the total volume of the ship.

Sketch a graph of the alien craft. How long is the ship?
What is the volume of the ship?
If the density of an average alien is approximately $3.5$ pounds per cubic foot, how much does the average alien weigh?

8-80.

One of the most popular events at the 1998 Winter Olympics in Nagano, Japan, was the luge. In this event, competitors lie on their backs and slide down an iced track on small sleds. At times, luge riders travel more than $120$ km per hour!

During one particular run, a competitor from Norway had the times and distances listed in the table below. The track is $1200$ m long. Enter the data points into your graphing calculator and then complete parts (a) through (c) below. Homework Help ✎

Determine the average velocity of the competitor during the run.
The graph of the distance can be modeled by $s(t)=$ $0.000145t^4-0.0246t^3+$ $1.315t^2-15.808t+$ $5.582$ using a quartic regression. Use your curve to approximate the velocity at $t = 96$ sec, when the athlete completed the race. Convert your result to km/hr. Is your result reasonable?
Graph the first derivative of your curve of best fit to represent the velocity. Explain its shape. What happens during the course of the run that helps explain its shape?

Time (sec)	Distance (m)
$0$ $20$ $40$ $50$ $70$ $90$ $96$	$0$ $70$ $210$ $375$ $415$ $780$ $1200$

8-81.

Given the following information about $f (x)$ , evaluate each integral. Homework Help ✎

$\int _ { a } ^ { b } f ( x ) d x = 2.5$

$\int _ { b } ^ { c } f ( x ) d x = - 5.0$

$\int _ { c } ^ { d } f ( x ) d x = 1.5$

$\int _ { c } ^ { a } f ( x ) d x$

$\int _ { a } ^ { c } 2 f ( x ) d x$

$\int _ { d } ^ { a } f ( x ) d x$

$\int _ { c } ^ { c } f ( x ) d x$

8-82.

Multiple Choice: The maximum value of $f(x) = x(x - 3)(x - 7)$ over the interval $0 ≤ x ≤ 7$ is nearest to: Homework Help ✎

$12$

$14$

$16$

$22$

8-83.

Multiple Choice: If $x^2 -2xy = 8$ , then when $x = 2$ , $\frac { d y } { d x }=$ Homework Help ✎

$-\frac { 1 } { 2 }$

$\frac { 1 } { 2 }$

$\frac { 3 } { 2 }$

$-\frac { 3 } { 2 }$

undefined

8-84.

Multiple Choice: The mean value of the function $f ( x ) = x \sqrt { 1 - x ^ { 2 } }$ over the interval $0 ≤ x ≤ 1$ is: Homework Help ✎

$\frac { 1 } { 3 }$

$\frac { 1 } { 6 }$

$\frac { 2 } { 3 }$

$\frac { 1 } { 2 }$

$\frac { 1 } { 4 }$

8-85.

Multiple Choice: $f(x) = e^{-x} - x^3 + 3x^2 - 10x$ changes concavity at $x =$ Homework Help ✎

$-2.03$

$-1.54$

$0.32$

$1.06$

$2.35$

8-86.

Multiple Choice: The graph of $y = g\left(x\right)$ shown below has horizontal tangents at $\left(–1, 1\right)$ and $\left(1, –1\right)$ . It has vertical tangent at $\left(0, 0\right)$ . For what values of $x$ on the open interval $\left(–2, 4\right)$ is $g$ not differentiable? Homework Help ✎

$2$ only

$0$ and $2$

$−1$ and $1$

$0$ only

$−1, 0, 1$ , and $2$

Piecewise labeled, g of x, left semicircle with vertices at (negative 2, comma 0), (negative 1, comma 1), & the origin, center semicircle, vertices at the origin, (1, comma negative 1), & (2, comma 0), left increasing concave up curve, starting at open point (2, comma negative 2), passing through the point (4, comma 4).

Calculus Third Edition