3. Volumes

Volume

• Let’s review how to find volume of a regular geometric prism. Whether it be a circular prism (cylinder), triangular prism, rectangular prism, etc., if we can find the area of the face, we need only to multiply by the distance between the two faces, h.

• In general V = A*h

Volume

V = A * h ( )V A x x ( )b

a

V A x dx

• The key will be finding the formula for the area of the base

Loaf of bread

• Imagine slicing a loaf of bread. It might look like this:

• At each slice, we have a face with a different area, but one that is a function of where along the loaf the slice was taken.

Loaf of bread• The measurable thickness of the slice, whether

it be for a sandwich or Texas Toast is ∆x. This thickness forms the distance, h, between the two faces. We can slice up the loaf from left to right using a uniform ∆x.

• Once again, if the cross sectional area of S in the plane through x and perpendicular to the x-axis is A(x) then the volume of S is

( )b

a

V A x dx

Example 1

• Show that the volume of a sphere of radius r is 34

3V r

Volume with known cross sections• Sometimes a shape is “built” by piling shapes

with a known cross sections perperdicular to an axis

• (see projects)• If the cross sections are perpendicular to the x-

axis, find the area in terms of x

• If the cross sections are perpendicular to the y-axis, find the area in terms of y

( )b

a

V A y dy

( )b

a

V A x dx

Cross Sections

• Problems will tell you what cross sections to use.

• The base of whatever figure it is will be found the same way we set up the integral to find area between 2 curves.

Area formulas for cross sections that show up quite a bit.

• In each of these cases, b is found as if we were finding the area between curves.

• http://mathdemos.org/mathdemos/sectionmethod/sectiongallery.html

2

2

2

Squares: Rectangles:

1 3Triangles: Equilateral Triangles: 2 4

Semicircles: 8

A b A bh

A bh A b

A b

Example 2

• Find the volume of a figure with base between the curves y=x+1 and y=x2-1 with the following cross sections perpendicular to x-axis.

Example 3

Example 4

• A region R, defined by the intersections of the graphs of y = 5x, y = -x/5 + 3, and y = 0 is the base of a solid whose cross section perpendicular to the y axis has area Find the volume of the solid.

2( ) 1A y y

Break!

Rotating a function around an axis• Many times a shape is obtained by rotating a

function around an axis. This is called a solid of revolution.

• http://curvebank.calstatela.edu/volrev/volrev.htm

• Whenever you rotate around an axis perpendicular to the slices, the cross-sections will be circular (DISCs)

• You can remember this by remembering perperDISCular

• The area of a circle is πr2. The radius is the value of the function

• To find volume, take integral of area of cross section

2b

a

r dx2( ( ))

b

a

f x dx

Example 5

• Find the volume of the solid formed by rotating the region bounded by the x-axis , and x=1 around the x-axis.

y x

Remember to always draw a picture!• Draw a picture of the region and draw the

representative slice. This makes the process much easier!

Example 6

• Find the volume of the solid formed by rotating the region bounded by y=1, , and x=0 around the line y=1.

y x

Example 7

• Find the volume of the solid formed by rotating the region bounded by y=8, , and x=0 around the y-axis.

3y x

What if the graph doesn’t reach the axis you are rotating around?

This makes the cross section a washer

R

r

2 2A R r 2 2( )A R r

Make sure you square each function separately,Don’t subtract first and then square!!

http://www.mathdemos.org/mathdemos/washermethod/gallery/gallery.html

Now there will be a void or hole

Example 8

• Find the volume of the region in the 1st quadrant obtained by rotating the graphs of y = cos x and y = sin x around the x-axis

Important things to consider

• Always draw a picture! Always draw a picture!• Draw R and r on the picture and label them with

an equation• When writing an equation for R and r, it will still

involve TOP – BOTTOM or RIGHT – LEFT. One of these in each case will be the axis of rotation itself.

• Don’t forget to square each radius before subtracting them! (Did I say that already?)

Example 9

• Find the volume of revolving the region enclosed by y = x and y=x2 around

Example 10

• Find the volume of the solid formed by rotating the region bounded by , y=0, x=0 and x=1 around the y-axis.

2 1y x

Can we do example 10 with a single integral?• Not if we slice perpendicular to the axis of

rotation, but what if we slice it more cleverly!• What if we sliced the region PARALLEL to the

axis of rotation. What would happen to a representative rectangle when taken for a spin around an axis parallel to it? Think “bundt cake”.

• http://www.mathdemos.org/mathdemos/shellmethod/gallery/gallery.html

• We call this a Cylindrical Shell.

How do we find the volume of this shell?• The shell, with finite thickness has two radii, but

as we slice thinner and thinner and thinner, the two radii approach each other. For an infinitely thin slice, we can use a single radius (think school-bought toilet paper)

• When the volume of solid is obtained by rotating a region paraSHELL to the axis of rotation, the volume is given by

2b

a

V rh2 (radius)(height of shell)

b

a

V dx 2 ( )( ( ))

b

a

V x f x dx

Method• Draw a rectangle perpendicular to the

axis of revolution – this will be the radius and is what will get revolved around axis

• Draw a line parallel to axis of revolution inside the area to be revolved – this will be the height and is the function

Example 11

• The region enclosed by the x-axis and the parabola is revolved around the line x=-1. What is its volume? What if we rotate around x=4 instead?

23y x x

Example 12

• Find the volume of the solid formed by revolving the region formed by about the y-axis. Use both vertical and horizontal slices. Compare your results.

2 and y x y x