8.2.3Disks, washers, or shells?

Using an Appropriate Method to Calculate Volume

8-87.

So far we have used disks, washers, and shells to calculate volumes of solids of revolution. How do you choose the appropriate method? While it is possible to use either the washer or shell method in many cases, one method might be easier than the other. It helps to have the following images in your mind when considering using a washer or a shell.

Very thin horizontal cylinder, with inner cylinder, center of all bases on the positive x axis, vertical rectangle labeled typical rectangle, between the cylinders, representing the difference in radii between the cylinders, and curved arrow showing rotation around x axis.

Vertical cylinder, slightly smaller inner cylinder, center of all bases on the positive y axis, vertical rectangle labeled typical rectangle, between the cylinders, representing the height of the cylinders, and curved arrow showing rotation around y axis.

Remember that in either case, the axis of revolution can be horizontal or vertical. For each part below:

Sketch the specified region.
Sketch an image of the rotated solid showing a typical slice.
Determine if you are using shells, disks, or washers to calculate the volume.
Set up the integral that will compute the volume.

The region between $x - y = -1$ and $y = x^2 - 6x + 7$ , rotated about the $y$ -axis.
The region between $x - y = -1$ and $y = x^2 - 6x + 7$ , rotated about the line $y = -3$ .
The region between $x = -y^2 + 3$ and $x = y^2 +1$ , rotated about the $y$ -axis.
The region between $x = -y^2 + 3$ and $x = y^2 +1$ , rotated about the line $y = 2$ .
The region between the $x$ - and $y$ -axes, $y = \cos(x) + 2$ , and $x = 2π$ , rotated about the $x$ -axis.
The region between the $x$ - and $y$ -axes, $y = \cos(x) + 2$ , and $x = 2π$ , rotated about the $y$ -axis.
The region between the $y$ -axis, $y = 2^x$ , and $y = -x + 6$ , rotated about the $x$ -axis.
The region between the $y$ -axis, $y = 2^x$ , and $y = -x + 6$ , rotated about the $y$ -axis

8-88.

Let $g(x)=\int_0^xf(t)dt$ where $y = f (t)$ is shown at right. Homework Help ✎

Evaluate $g(2)$ and $g(4)$ .
Is $g ( - 2 ) = \int _ { - 2 } ^ { 0 } f ( t ) d t$ ?
Express $\int _ { - 2 } ^ { 2 } f ( t ) d t$ in terms of $g$ .
Is $g$ differentiable over the interval $−2 < x < 6$ ? Explain.
Determine all values of $x$ in the interval $−2 < x < 6$ where $g$ has a relative maximum.
Write the equation of the line tangent to $g$ at $x = 4$ .
Determine all values of $x$ in the interval $−2 < x < 6$ where $g$ has a point of inflection.

Continuous Piecewise labeled f of x, left segment from (negative 2, comma 0), to (0, comma 2), center segment from (0, comma 2) to (2, comma 0), right semicircle, vertices at the points (2, comma 0), (4, comma negative 2), & (6, comma 0).

8-89.

Examine the derivatives below. Consider the multiple tools available for finding derivatives and use the best strategy for each part. Evaluate each derivative and briefly describe your method. Homework Help ✎

$\frac { d } { d y } [ \frac { e ^ { y } } { y } ]$

$\frac { d } { d x } [ \sqrt { \operatorname { sin } ( x ) } ]$

$\frac { d } { d \theta } [ 5 \operatorname { cos } ^ { 2 } ( \theta ) + 5 \operatorname { sin } ^ { 2 } ( \theta ) ]$

8-90.

Multiple Choice: $\lim \limits _ { h \rightarrow 0 } \frac { ( x + h ) ^ { 1 / 2 } - x ^ { 1 / 2 } } { h } =$ Homework Help ✎

$\sqrt { x }$

$\frac { 1 } { \sqrt { x } }$

$\frac { \sqrt { x } } { 2 }$

$\frac { 1 } { 2 \sqrt { x } }$

$\frac { x ^ { 2 } } { 2 }$

8-91.

Multiple Choice: The expression $\frac { 1 } { 20 } [ ( \frac { 1 } { 20 } ) ^ { 2 } + ( \frac { 2 } { 20 } ) ^ { 2 } + ( \frac { 3 } { 20 } ) ^ { 2 } + \ldots + ( \frac { 20 } { 20 } ) ^ { 2 } ]$ is a Riemann sum approximation of: Homework Help ✎

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

$\frac { 1 } { 20 } \int _ { 0 } ^ { 1 } x ^ { 2 } d x$

$\frac { 1 } { 20 } \int _ { 0 } ^ { 1 } ( \frac { x } { 20 } ) ^ { 2 } d x$

$\int _ { 0 } ^ { 1 } ( \frac { x } { 20 } ) ^ { 2 } d x$

$\frac { 1 } { 20 } \int _ { 1 } ^ { 20 } ( \frac { x } { 20 } ) ^ { 2 } d x$

8-92.

Multiple Choice: What is the area of the region in the first quadrant bounded by the graphs of $y = e^x$ and $y = -x + 3$ ? Homework Help ✎

$0.726$

$0.782$

$0.855$

$1.208$

$2.063$

8-93.

Multiple Choice: The graph of $y = f ^\prime(x)$ is shown at right. Which of the graphs below could be the graph of $y = f (x)$ ? Homework Help ✎

Upward parabola labeled, f prime of x, vertex at the point (1, comma negative 1, passing through the points (2, comma 0), & the origin.

8-94.

Multiple Choice: If $f ( x ) = \frac { e ^ { \sqrt { x } } } { \sqrt { x } }$ , then $f^\prime(x) =$ Homework Help ✎

$e ^ { \sqrt { x } }$

$\frac { e ^ { \sqrt { x } } } { x }$

$\frac { e ^ { \sqrt { x } } ( x - \sqrt { x } ) } { 2 x ^ { 2 } }$

8-95.

Multiple Choice: If $\int _ { a } ^ { b } f ( x ) d x = 2 b - a$ , then $\int _ { a } ^ { b } ( f ( x ) - 3 ) d x =$ Homework Help ✎

$2b - a - 3$

$2b − a$

$2a − b$

$4a − b$

$3a − 6b$

8-96.

Multiple Choice: The function $f (x) = x^2$ is bounded by the $y$ -axis and the line $y = 4$ . The volume of the solid generated by revolving the region about the $y$ -axis can be computed by which of the following integrals? Homework Help ✎

$2 \pi \int _ { 0 } ^ { 2 } x ( 4 - x ^ { 2 } ) d x$

$\pi \int _ { 0 } ^ { 4 } y d y$

$\pi \int _ { 0 } ^ { 4 } \sqrt { y } d y$

A. I only

B. II only

C. III only

D. I and II

E. I and III

Calculus Third Edition