8.1.3How do I remove the hole?

The Washer Method

8-25.

A region is bounded by the $y$ -axis and the lines $y = x$ and $y = 4$ .

Describe the solid that is formed when the region is revolved about the $x$ -axis. Using geometry, calculate the volume of the solid.
Calculus can be used to determine the volume of a solid of revolution by looking at a typical slice of the solid. Again, start by looking at the typical rectangle and revolve it around the $x$ -axis to form the typical disk, as shown in the diagrams below. When the disk has a hole in the center, it is referred to as a washer. Write an expression for the volume of the “typical” washer.
Write and evaluate the integral that accumulates the volume of these washers.

8-26.

Sketch each region described by the boundaries below. Rotate each region about the indicated axis. Decide if horizontal or vertical rectangles should be used to calculate each volume. Then, set up and evaluate an integral to determine the volume of each solid of revolution. Check your solution with your calculator.

$y = x + 3$ , $y = (x - 3)^2$ , $x = 0$
Rotated about the $y$ -axis.

$y = x + 3$ , $y = (x - 3)^2$ , $x = 0$
Rotated about the $x$ -axis.

8-27.

2 downward parabola, top labeled y = negative 2 x squared + 2, vertex at the point (0, comma 2), starting @ (negative 1, comma 0) & ending @ (1, comma 0), bottom parabola labeled y = negative x squared + 1, vertex at the point (0, comma 1), starting at (negative 1, comma 0), & ending @ (1, comma 1), with shaded region between the parabolas. While watching Asteroid Journey one evening, Sprock drew a picture, which resembled the Droidfleet emblem shown at right.

If the region is rotated about the $x$ -axis and then sliced vertically, draw a picture of the rotated solid with a typical washer drawn in. To which axis is the washer perpendicular?
Write an expression for the volume of a typical washer.
Use your volume expression to set up an integral representing the total volume of the solid.
How can you use symmetry to write an integral giving the same volume, but easier to integrate? What is the integral?

8-28.

Sketch each region contained by the boundaries below. Decide if you should use horizontal or vertical rectangles to calculate each area. Then, set up and evaluate an integral to calculate the area. Check your solution with your calculator. Homework Help ✎

$y =\sqrt { x }$ , $y = -x$ , $x = 1$ , and $x = 4$
$x + y = 2$ and $x = -y^2 + 4$

8-29.

Set up and evaluate an integral to calculate the area enclosed in the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 9 } = 1$ . Homework Help ✎

8-30.

Set up and evaluate an integral to calculate the volume of the ellipsoid that is formed by revolving the ellipse in problem 8-29 about the $x$ -axis. Homework Help ✎

8-31.

Determine the value of $h$ so that each of the functions below is continuous. Homework Help ✎

$f ( x ) = \left\{ \begin{array} { c l } { h } & { \text { for } x = 3 } \\ { \frac { x ^ { 2 } - x - 6 } { x - 3 } } & { \text { for } x \neq 3 } \end{array} \right.$

$f ( x ) = \left\{ \begin{array} { c c } { \sqrt { x + h } } & { \text { for } x \leq 3 } \\ { - x + h } & { \text { for } x > 3 } \end{array} \right.$

Determine where the functions in parts (a) and (b) are differentiable.

8-32.

No calculator! Integrate. Homework Help ✎

Compute without a calculator

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

$\frac { d } { d x } \int _ { 4 } ^ { x ^ { 2 } } f ( x ) d x$

$\int \frac { \operatorname { sec } ( x ) \operatorname { tan } ( x ) } { 1 + \operatorname { sec } ^ { 2 } ( x ) } d x$

$\int \frac { x } { \sqrt { 1 - x ^ { 4 } } } d x$

$\int _ { 0 } ^ { \pi / 2 } \operatorname { tan } ( \frac { x } { 2 } ) d x$

8-33.

No calculator! Given $f ( x ) = \frac { 1 - x } { x - 3 }$ , determine all intervals where $y = f (x)$ is concave up. Homework Help ✎

Compute without a calculator

8-34.

Set up an integral that can be used to calculate the volume of the solid obtained by revolving the region defined below about the $y$ -axis. Homework Help ✎

$x \leq \sqrt { 25 - y ^ { 2 } } \text { and } x \geq 3$

8-35.

Is there a $c$ in the interval $[0, 1]$ for $f (x) = x^2 - x$ where $f ^\prime(c) = 0$ ? Justify your answer. Homework Help ✎

Types of Slices (OR “CROSS-SECTIONS”)

When calculating a volume by slicing, several different shapes of slices can be created. Below are the common names for these slices.

DISKS:

Vertical cylinder, with very thin height, top base shaded. Small cylinders stacked together, such as in the Lime Lab. The volume of each disk can be found with the cylinder formula $V = πr^2h$ , where $r$ is the radius and $h$ is usually $Δx$ or $Δy$ .

WASHERS:

2 concentric cylinders, with very thin heights, with region between the cylinders shaded. These are disks with holes in the middle, such as those in problem 8-25. The volume of each washer can be found using $V = π(R^2 - r^2)h$ where $R$ is the outer radius, $r$ is the inner radius, and $h$ is usually $Δx$ or $Δy$ .

PRISMS:

Rectangular prism, with very thin height, top base shaded. This general name applies to all other slices that have non-circular cross-sections, such as the Goodslice bread example. The base of each prism can be triangular, square, rectangular, trapezoidal, etc. The volume of each prism is generally found by using $V = (\text{base area})(\text{height})$ where height is usually $Δx$ or $Δy$ .

8-36.

Linda has enrolled in a ceramics course and is concerned about having enough clay to complete her projects, listed below, so she decides to calculate the volume of each object she will create by slicing. Which direction should she slice, and what type of cross-section should she use to calculate the volume of each of her projects? Homework Help ✎

A vase.

A pyramid.

A rolling pin.

Calculus Third Edition

Types of Slices (OR “CROSS-SECTIONS”)