Volumes of Solids with Known Cross Sections
Volumes of solids with known cross sections is a fairly easy concept to grasp. It is similar to the idea of the volume of solids of rotation in that in both types of situations, there is a region on a graph bounded by a series of lines, and the volume of solids with known cross sections is essentially the same concept. In equations involving rotation, the equation is:

The reason that there is all of the mumbo-jumbo with pi and the radius nonsense is that, since the object is being rotated, the cross sections it is forming are circles, and, essentially, when finding the volume, all you are doing is multiplying cross sections by the changes in x to get a 3D solid, and then adding up all of this tiny 3D solids in order to find what the one large volume is. Therefore, when the cross section is known, the equation turns out to be: A(x) is the area of one of these cross sections
dx is an infinitely small value of the change in x
The limits of integration are simply the x-values that bind the region of the graph that represents the base of the solid
(Note: There is no pi in this equation because there is no rotating around, so no circles are formed)
(Additional note: the A(x) is for when the cross section is perpendicular to the x-axis. A(y) is used if the cross section is perpendicular to the y-axis.)
The cross section can be any sort of shape, and the area that is plugged into the equation and the way in which you use the graph depends on whether or not the cross section is perpendicular to the x-axis or the y-axis.

EXAMPLES: Find the volume of a solid with a base bounded by the equations and y=1 and the x=4 if the cross sections perpendicular to the x-axis are squares.
First, it is wise to get a picture of the graph.

Then, once you know what the graph looks like, you can visualize the different cross sections

(Those are supposed to be squares, but you can't expect much from microsoft paint)
Anyways, in order to find the volume, you must know how to find the area of your cross section. Our cross section is a square with a base perpendicular to the x-axis, and the area of a square is simply A=
X2 and, in our case, the side is equal to (R-r), which is , so our area is .

Then, one can plug this area into our equation , knowing that the limits of integration are x=1 and x=4. Our equation becomes:

This can be easily plugged into the calculator, or worked out so that it becomes , which becomes
(1/2)x2-(4/3)x3/2+x from 1 to 4, which becomes
[(1/2)(4)2-(4/3)(4)3/2+4]-[
(1/2)(1)2-(4/3)(1)3/2+1], which becomes [8-(32/3)+4]-[.5-(4/3)+1], which becomes (4/3)-(1/6), which is 7/6.

Example 2: A solid has its base is the region bounded by the lines x + y = 4, x = 0 and y = 0 and the cross section is perpendicular to the x-axis are equilateral triangles. Find its volume. Solution : The cross section is an equilateral triangle, perpendicular to the x-axis. Its side has its ends on the line x + y = 4 and the x-axis. \ Side (s) = 4 - x

Here's a Youtube video that might be able to help any who still don't quite understand. It is actually a really good, straightforward video. In fact, you could probably watch this video and skip the entire rest of the page, but don't do that.

Volumes of solids with known cross sections is a fairly easy concept to grasp. It is similar to the idea of the volume of solids of rotation in that in both types of situations, there is a region on a graph bounded by a series of lines, and the volume of solids with known cross sections is essentially the same concept. In equations involving rotation, the equation is:

The reason that there is all of the mumbo-jumbo with pi and the radius nonsense is that, since the object is being rotated, the cross sections it is forming are circles, and, essentially, when finding the volume, all you are doing is multiplying cross sections by the changes in x to get a 3D solid, and then adding up all of this tiny 3D solids in order to find what the one large volume is. Therefore, when the cross section is known, the equation turns out to be:

A(x) is the area of one of these cross sections

dx is an infinitely small value of the change in x

The limits of integration are simply the x-values that bind the region of the graph that represents the base of the solid

(Note: There is no pi in this equation because there is no rotating around, so no circles are formed)

(Additional note: the A(x) is for when the cross section is perpendicular to the x-axis. A(y) is used if the cross section is perpendicular to the y-axis.)

The cross section can be any sort of shape, and the area that is plugged into the equation and the way in which you use the graph depends on whether or not the cross section is perpendicular to the x-axis or the y-axis.

EXAMPLES: Find the volume of a solid with a base bounded by the equations and y=1 and the x=4 if the cross sections perpendicular to the x-axis are squares.

First, it is wise to get a picture of the graph.

Then, once you know what the graph looks like, you can visualize the different cross sections

(Those are supposed to be squares, but you can't expect much from microsoft paint)

Anyways, in order to find the volume, you must know how to find the area of your cross section. Our cross section is a square with a base perpendicular to the x-axis, and the area of a square is simply A=

X2 and, in our case, the side is equal to (R-r), which is , so our area is .

Then, one can plug this area into our equation , knowing that the limits of integration are x=1 and x=4. Our equation becomes:

This can be easily plugged into the calculator, or worked out so that it becomes , which becomes

(1/2)x2-(4/3)x3/2+x from 1 to 4, which becomes

[(1/2)(4)2-(4/3)(4)3/2+4]-[

(1/2)(1)2-(4/3)(1)3/2+1], which becomes [8-(32/3)+4]-[.5-(4/3)+1], which becomes (4/3)-(1/6), which is

7/6.Example 2: A solid has its base is the region bounded by the lines x + y = 4, x = 0 and y = 0 and the cross section is perpendicular to the x-axis are equilateral triangles. Find its volume.

Solution :The cross section is an equilateral triangle, perpendicular to the x-axis. Its side has its ends on the line x + y = 4 and the x-axis.\ Side (s) = 4 - x

A (x) = area of the equilateral triangle

\ The volume of the solid is given by

Here's a Youtube video that might be able to help any who still don't quite understand. It is actually a really good, straightforward video. In fact, you could probably watch this video and skip the entire rest of the page, but don't do that.

Sources:

The textbook

My mind

Microsoft Paint

Youtube

Suchana's page for the rotated solid equation

http://www.sitmo.com/latex/

http://www.ucl.ac.uk/Mathematics/geomath/level1/diffnb/diffpic10gr1.gif

http://www.pinkmonkey.com/studyguides/subjects/calc/chap8/c0808302.asp

http://www.cliffsnotes.com/study_guide/Volumes-of-Solids-with-Known-Cross-Sections.topicArticleId-39909,articleId-39906.html