8.1.6Which method should I use?

Disk and Washer Problems

8-54.

The region at right is bounded by the $x$ -axis and $f ( x ) = \sqrt { 1 - x ^ { 2 } }$ . Describe the solid formed when the region is revolved around the line $y = 2$ . Then calculate the volume of the solid.

Horizontal dashed line at y = 2, top half of circle, centered at the origin, with diameter endpoints at (negative 1, comma 0), & (1, comma 0), shaded region inside semi circle & above x axis.

8-55.

When each shaded region below is rotated about the given axis, decide if it is best to slice the resulting solid horizontally or vertically in order to calculate the volume. Describe the shape of each slice. Also indicate if the slices will have thickness $dx$ or $dy$ . Finally, set up an integral for the volume.

Rotated about the $x$ -axis.

Rotated about the line $y = −1$ .

Rotated about the $y$ -axis.

Rotated about the line $x = 5$ .

8-56.

Jasmine has a bud vase with a very narrow neck. She has a tendency to overfill the vase because the water rises so quickly up the neck. To prevent future spills, she wants to construct a cylindrical tube that is the same height and volume as the vase.

Using her excellent analytical skills, she has found that the volume can be generated by rotating a region bounded by the y-axis and the curve $x = \frac { 6 } { y } - \frac { 5 } { y ^ { 2 } }$ from $y = 1$ to $y = 13$ . (The units for $x$ and $y$ are in inches.)

Calculate the volume of the vase. Include a diagram with a typical slice. What quantity represents the thickness of the slice?
Using your results, determine the dimensions for the cylindrical tube that she wants to construct.

8-57.

GEOMETRY PROOF: VOLUME OF A CONE

A previous geometry class introduced a formula often used to calculate the volume of a cone. However, an integral can also be used to calculate the volume of a cone. In order to apply calculus, view a cone as a right triangle revolved about the $y$ -axis.

If the cone has height $h$ and base radius $r$ , write an equation for the line that will form the triangle. (Hint: The $x$ -intercept is $(r, 0)$ and $y$ -intercept is $(0, h)$ .)
Set up an integral that will accumulate the volume of the disks.
Evaluate your integral from part (b). Remember that $r$ and $h$ are constants. Did you get $V =\frac { 1 } { 3 }π r^2h$ ?

Shaded right triangle with vertices at the origin, (0, comma h), & (r, comma 0), dashed segment from (0, comma h) to (negative r, comma 0), a rectangle half way up the positive y axis, with midpoints of left & right sides on the slanted segments, a thin cylinder on the side with arrow pointing to the rectangle.

8-58.

Suppose the region in problem 8-56 is revolved around the line $x = -1$ instead of the $y$ -axis. Set up and evaluate a new integral that will calculate the volume of the solid. Homework Help ✎

8-59.

No calculator! Integrate. Homework Help ✎

$\int _ { 0 } ^ { 8 } | x - 4 | d x$

$\int w ( w ^ { 2 } - 1 ) d w$

$\int _ { - 1 } ^ { - 5 } - 7 d x$

$\int _ { 0 } ^ { 5 } \sqrt { 25 - x ^ { 2 } } d x$

Compute without a calculator

8-60.

THE MATRYOSHKA DOLL

For generations, special wooden nesting dolls called matryoska have been handcrafted in Russia. This type of cylindrical doll opens at the midsection and nested inside is a smaller wooden doll (with the same shape) which itself opens to another doll. Some matryoshka are made up of ten or more dolls! The smallest doll, often less than a centimeter tall, is solid wood.

Explain why, under ideal conditions, the sum of the volumes of the nested wooden shells of the different-sized dolls will calculate the full volume of the interior of the largest doll. Homework Help ✎

8-61.

Instead of using an area formula, Joshua wants to use calculus to find the area of a circle. He realizes that one way to shade a circle is to draw thin concentric circles from the center out, so close together that they entirely cover the circle.

To calculate the area, he needs to add up the area covered by each “circumference” of these circles. If the radius starts at a value of $0$ and ends at a value of $r$ , use an integral to prove that the sum of these circumferences equals the area of the circle. Homework Help ✎

7 concentric circles, of uneven radii, every other one shaded, arrow from biggest shaded circle points to rectangle, left side labeled, d x, bottom side labeled, 2 pi x.

Math Notes

Accumulating Area and Volume With Integrals

Riemann sums helped add the areas of sequential rectangles to determine an approximate total area of a region. By taking the limit as the number of rectangles approaches infinity, we get an accurate measure of area.

Likewise, if we add sequential volumes of disks or other slices, we obtain an approximate total volume of a given solid. By taking the limit as the number of slices approaches infinity, we get the exact volume.

We can skip the limiting process if we think of “infinitely thin” rectangles or slices from the start. For example, in problem 8-61, Joshua added “infinitely thin circumferences” with an integral to determine the area.

If the area of an infinitely thin slice of a solid at an arbitrary $x$ -value is $A(x)$ and its thickness is $dx$ , then $A(x)dx$ is the volume of that infinitely thin slice. Adding up these slices as $x$ changes from $a$ to $b$ will accumulate the total volume. In symbols:

$\text{volume}= \int_a^b A(x)dx$

8-62.

GEOMETRY PROOF: VOLUME OF A SPHERE

Another way to view a sphere is as a semicircle rotated about an axis. Use a generic semicircle such as $y = \sqrt { r ^ { 2 } - x ^ { 2 } }$ to prove that the volume of a sphere with radius $r$ is $\frac{4}{3}\pi r^3$ . Homework Help ✎

8-63.

The diagram at right shows the region bounded by the $x$ -axis, $f(x) = 0.5x^2$ , $x = 1$ , and $x = 3$ . The region is revolved about the $y$ -axis to create the solid shown with dotted lines. 8-63 HW eTool. Homework Help ✎

Describe a method you can use to determine the volume of the solid.
Set up and evaluate the integrals needed to calculate the volume. (Using washers, the solution will require two integrals.)

8-64.

At which point(s) on the graph of $y = f(x)$ at right, do the following situations occur? Homework Help ✎

$f^\prime(x)$ and $f^{\prime\prime}(x)$ are both positive
$f^\prime(x)$ and $f^{\prime\prime}(x)$ are both negative
$f^\prime(x)$ is positive and $f^{\prime\prime}(x)$ is negative
$f^\prime(x)$ is negative and $f^{\prime\prime}(x)$ is positive

Calculus Third Edition

Accumulating Area and Volume With Integrals