8.3.3What shape is the cross-section?

Cross-Section Problems

8-120.

Your teacher will provide you with a model. So far you have examined several volumes that are formed by revolving a region about a line. In each of these cases, the cross-sections were circular. The same summation process can be applied to solids with non-circular cross-sections.

For example, consider a corner cut from a box where each cross-section forms an isosceles right triangle as shown in the diagram at right. Given base edges of $12$ cm and a height of $4$ cm, a line can determine the dimensions of the slice. Since we are slicing perpendicular to the $x$ -axis, the thickness of a typical slice is $dx$ .

Write the equation of the line that will determine the dimensions of the typical slice.
What is the volume of a typical slice?
Set up and evaluate an integral that will calculate the volume of this solid.

8-121.

Set up and evaluate an integral to calculate the volume of a solid with a base that is the bounded region and cross-sections perpendicular to the indicated axis. Be sure to draw a diagram showing a typical slice.

The region bounded by $y =\sqrt { x }$ , $y = 0$ , and $x = 4$ with semicircular cross-sections perpendicular to the $x$ -axis.
The region bounded by $y = \sqrt { 9 - x ^ { 2 } }$ , $y = 0$ , and $x = 0$ with rectangular cross-sections (height is half the length of the base) perpendicular to the $x$ -axis.
The region bounded by $y =|x|$ and $y = 5$ with right triangular cross-sections (height is twice the length of the base) perpendicular to the $y$ -axis.
The region bounded by $y = 2^x$ and $y =\sqrt { x }+1$ with semicircular cross-sections perpendicular to the $x$ -axis.
The region bounded by $y=\frac { 1 } { 4 }x^2 + 4$ , $y = 2x$ and $x = 0$ with square cross-sections perpendicular to the $y$ -axis.

8-122.

Your teacher will provide you with a model. The town council of Smellyville is concerned that the city dump is not sufficient for the needs of the community. They have proposed creating a second landfill but have met with strong opposition from a citizens group near the proposed site. The group claims a second site is not necessary for the town. The council stated that they would delay construction on the site if the current landfill is still available in $5$ years.

The current dump has a square base with length of $100$ yards. The sides of the dump are formed by the parabolas $y = 0.05(x ± 50)^2$ , where $x$ and $y$ are in yards. Each cross-section is a square. Currently, the dump is filled to a height of $12$ yards, but more trash is on its way!

Write an expression, in terms of a single variable, that will calculate the volume of a typical slice.
How much garbage is already in the dump?
When full, the dump will have a depth of $20$ yards. How much room is left in the dump?
If $60$ cubic yards of garbage arrive at the dump every day, how many years will it be before the dump is full? Will construction be delayed?

8-123.

GEOMETRY PROOF: VOLUME OF A SQUARE-BASED PYRAMID

Think of a pyramid as a stack of square slices with decreasing side lengths, similar to the diagram at right.

Sketch a generic pyramid so that the $y$ -axis lies on the altitude (or “height”), as shown in the diagram at right. Label the length of the base $b$ and the height $h$ .
Write the general equation of one side of the pyramid (the bold line) that will help define the width of each square cross-section.
Set up and evaluate an integral that will add up the volumes of the square slices that form the pyramid. Remember that $b$ and $h$ are constants. Did you get $\frac { 1 } { 3 }b^2h$ ?

8-124.

You have designed a model of a square-based pyramid in honor of your calculus teacher. The height will be $10$ inches. The equation of one side of the pyramid is: $f(x) = -0.5x + 10$ . 6-124 HW eTool. Homework Help ✎

Sketch a diagram of your pyramid. A complete diagram includes the function, the $x$ - and $y$ -axes, and a typical slice labeled with the appropriate dimensions.
Set up and evaluate an integral that calculates the exact volume of your pyramid.

8-125.

Determine the point of intersection of the two lines tangent to $y = \frac { 1 } { 1 + x ^ { 2 } }$ at $x = -1$ and $x = 2$ . Homework Help ✎

8-126.

Let $f ( x ) = \sqrt { x - 4 }$ . Homework Help ✎

Calculate the area of the region bounded by $y = f(x)$ , the $x$ -axis, and $x = 8$ .
If the line $x = c$ divides the region from part (a) into two pieces of equal area, what is the value of $c$ ?
Calculate the volume of the solid that is formed by rotating the region described in part (a) about the $x$ -axis.
If a plane perpendicular to the $x$ -axis at $d = x$ divides the solid in part (c) into two parts of equal volume what is the value of $d$ ?

8-127.

Multiple Choice: The base of a solid is a region in the first quadrant under the curve $f ( x ) = \sqrt { 4 - x }$ , as shown in the diagram at right. The cross-sections of the solid are squares perpendicular to the $x$ -axis. What is the volume of the solid? Homework Help ✎

$\frac { 16 } { 3 }$

$\frac { 64 } { 9 }$

$8π$

$\frac { 16 \pi } { 3 }$

First quadrant decreasing concave down curve, starting at the point (0 comma 2), stopping at the point (4, comma 0), shaded region under the curve, above x axis, & right of y axis.

8-128.

Multiple Choice: If $x^2 + y^2 = 25$ then $\frac { d ^ { 2 } y } { d x ^ { 2 } }=$ Homework Help ✎

$- \frac { x } { y }$

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

$- \frac { 25 } { y ^ { 3 } }$

$\frac { x ^ { 2 } - y ^ { 2 } } { y ^ { 3 } }$

$\frac { y ^ { 2 } - x } { y ^ { 3 } }$

8-129.

Multiple Choice: Given $f ( x ) = \operatorname { ln } | 4 - x ^ { 2 } |$ , $f^\prime(x)$ is: Homework Help ✎

$| \frac { - 2 x } { 4 - x ^ { 2 } } |$

$| \frac { 2 x } { 4 - x ^ { 2 } } |$

$\frac { 2 x } { | x ^ { 2 } - 4 | }$

$\frac { 2 x } { x ^ { 2 } - 4 }$

$\frac { 1 } { | x - 2 | }$

8-130.

Multiple Choice: Given $f(x)=\frac { 1 } { x }- 4x^2 + 7x$ , then $f$ is concave up for: Homework Help ✎

$x ≠ 0$

no values of $x$

$x < 0$

$x < 0 \text { or } x > \frac { 1 } { \sqrt [ 3 ] { 4 } }$

$0 < x < \frac { 1 } { \sqrt [ 3 ] { 4 } }$

8-131.

Multiple Choice: If the slope of $y = f (x)$ is defined by $\frac { d y } { d x } = \frac { 2 x - 3 } { 4 y }$ where $f(1)=-2$ , then $f(2)$ is: Homework Help ✎

$-1$

$−2$

undefined

8-132.

Multiple Choice: The equation of the line normal to the curve $y = x^4 + 3x^3 + 2$ at the point where $x = 0$ is: Homework Help ✎

$y = x$

$y = 13x$

$y = 0$

$y = x + 2$

$x = 0$

Calculus Third Edition