8.1.1How do I slice it?

Volumes by Slicing

8-1.

LIME LAB, Part One

Simone has a part time job working as a waitress at a fine restaurant. While she enjoys her job, her thoughts are frequently directed towards her goal of becoming an engineer. One day, she looks at the sliced limes that are placed on each water glass and realizes that the slices can be used to estimate the volume of an entire lime. She promptly slices up a lime, creating cross-sections.

What is the geometric term that best describes the shape of each cross-section in this situation?
Using a similar slicing process, determine the volume of the lime provided by your teacher. Be sure to note the dimensions, units, and the method that you used.
Simone knows that by using the symmetry of the lime, a model of the lime can be generated by rotating a “flag.” Sketch the flag that will generate the shape of your lime when rotated.

8-2.

LIME LAB, Part Two

The volume of the lime in problem 8-1 can be approximated with the volume of each slice.

By adding the volumes of successive slices, a Riemann sum is formed. Explain how the expression below represents this sum.
$\displaystyle\sum _ { i = 1 } ^ { n }\text{area of the } i^{\text{th}} \text{slice} (\text{thickness})$
Explain how you can get an exact volume. Write an expression that will accomplish this.

8-3.

Curved vase with the following diameters from bottom to the top, 11.8, 14.3, 16.5, 18, 19, 18.1, 15.3, 11, 8.4, & 7. Starting at the bottom of a rare vase, Warren measures the diameter (in cm) every $5$ cm up the vase. He makes a total of ten measurements, shown at right.

Use his data to estimate the volume of the vase.
If Warren instead takes measurements every $2$ cm, what will happen to the estimate?
Continuing this process, describe how Warren should go about determining the exact volume of the vase.

8-4.

Remember that $\lim\limits _ { n \rightarrow \infty } \displaystyle\sum _ { i = 0 } ^ { n - 1 } f ( a + ( \Delta x ) i ) \Delta x = \int _ { a } ^ { b } f ( x ) d x$ . Homework Help ✎

What does " $dx$ " by itself represent?
What does the product $f(x)dx$ represent?

First quadrant graph, increasing curve opening up, starting about 1 fourth up on y axis, & 3 vertical shaded bars of equal width, bottom edges on x axis, left edge of first bar labeled, a, right edge of last bar labeled, b, top left vertex of each bar, on the curve.

8-5.

Sketch the region bounded by the curve $f(x) =\sqrt { x }$ the $x$ -axis for $4 ≤ x ≤ 9$ . Homework Help ✎

Draw a typical rectangle in the region and describe how the height and width are determined.
Set up and evaluate an integral to calculate the area of the region.

8-6.

Compute the following derivatives. Homework Help ✎

$\frac { d } { d x } \sqrt [ 3 ] { 5 x ^ { 2 } - 17 x }$

$\frac { d } { d x } ( ( 4 x ^ { 3 } + 2 x - 5 ) ( 8 - 7 x ^ { 3 } ) )$

$\frac { d } { d x } \operatorname { sin } ( \operatorname { sin } ( t ) )$

$\frac { d } { d x } \int _ { 7 x } ^ { \operatorname { cos } ( x ) } \operatorname { ln } ( t ^ { 2 } ) d t$

8-7.

Evaluate $\int _ { - 1 } ^ { 1 } [ \pi ( - 2 x ^ { 2 } + 2 ) ^ { 2 } - \pi ( - x ^ { 2 } + 1 ) ^ { 2 } ] d x$ . Homework Help ✎

8-8.

No calculator! Evaluate the following integrals. Homework Help ✎

$\int \frac { d } { d x } ( e ^ { 3 x } \operatorname { sin } ( 5 x ) ) d x$

$\int _ { 3 } ^ { 5 } e ^ { 4 x + 2 } d x$

$\int \frac { \operatorname { sec } ( x ) \operatorname { sin } ( x ) } { \operatorname { cos } ( x ) } d x$

$\int \operatorname { sin } ( 2 x ) \cdot 2 ^ { \operatorname { sin } ^ { 2 } ( x ) } d x$

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

Compute without a calculator

8-9.

The value of a rare stamp was $$250$ in 1989. The stamp has increased in value by $12\%$ each year. Homework Help ✎

What was the stamp worth in the year 2000?
What was the average value of the stamp from 1989 to 2000?

8-10.

Use the graph of $y = g(x)$ at right to answer the following questions. Provide reasons for each value found. Homework Help ✎

For what values of $x$ is $g(x)$ undefined?
For what values of $x$ is $g\left(x\right)$ discontinuous?
For what values of $x$ is $g(x)$ non-differentiable?

8-11.

Determine the following limits for $y = g(x)$ graphed in the previous problem. Homework Help ✎

$\lim\limits _ { x \rightarrow - 4 ^ { + } } g ( x )$

$\lim\limits _ { x \rightarrow - 4 ^ { - } } g ( x )$

$\lim\limits _ { x \rightarrow 2 } g ( x )$

$\lim\limits _ { x \rightarrow 1 } g ( x )$

Calculus Third Edition