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Rather than looking at an example of the washer method with the y -axis as the axis of revolution, we now consider an example in which the axis of revolution is a line other than one of the two coordinate axes. The same general method applies, but you may have to visualize just how to describe the cross-sectional area of the volume.

The washer method with a different axis of revolution

Find the volume of a solid of revolution formed by revolving the region bounded above by f ( x ) = 4 x and below by the x -axis over the interval [ 0 , 4 ] around the line y = −2 .

The graph of the region and the solid of revolution are shown in the following figure.

This figure has two graphs. The first graph is labeled “a” and has the two curves f(x)=4-x and -2. There is a shaded region making a triangle bounded by the decreasing line f(x), the y-axis and the x-axis. The second graph is the same two curves. There is a solid formed by rotating the shaded region from the first graph around the line y=-2. There is a hollow cylinder inside of the solid represented by the lines y=-2 and y=-4.
(a) The region between the graph of the function f ( x ) = 4 x and the x -axis over the interval [ 0 , 4 ] . (b) Revolving the region about the line y = −2 generates a solid of revolution with a cylindrical hole through its middle.

We can’t apply the volume formula to this problem directly because the axis of revolution is not one of the coordinate axes. However, we still know that the area of the cross-section is the area of the outer circle less the area of the inner circle. Looking at the graph of the function, we see the radius of the outer circle is given by f ( x ) + 2 , which simplifies to

f ( x ) + 2 = ( 4 x ) + 2 = 6 x .

The radius of the inner circle is g ( x ) = 2 . Therefore, we have

V = 0 4 π [ ( 6 x ) 2 ( 2 ) 2 ] d x = π 0 4 ( x 2 12 x + 32 ) d x = π [ x 3 3 6 x 2 + 32 x ] | 0 4 = 160 π 3 units 3 .
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Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of f ( x ) = x + 2 and below by the x -axis over the interval [ 0 , 3 ] around the line y = −1 .

60 π units 3

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Key concepts

  • Definite integrals can be used to find the volumes of solids. Using the slicing method, we can find a volume by integrating the cross-sectional area.
  • For solids of revolution, the volume slices are often disks and the cross-sections are circles. The method of disks involves applying the method of slicing in the particular case in which the cross-sections are circles, and using the formula for the area of a circle.
  • If a solid of revolution has a cavity in the center, the volume slices are washers. With the method of washers, the area of the inner circle is subtracted from the area of the outer circle before integrating.

Key equations

  • Disk Method along the x -axis
    V = a b π [ f ( x ) ] 2 d x
  • Disk Method along the y -axis
    V = c d π [ g ( y ) ] 2 d y
  • Washer Method
    V = a b π [ ( f ( x ) ) 2 ( g ( x ) ) 2 ] d x

Derive the formula for the volume of a sphere using the slicing method.

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Use the slicing method to derive the formula for the volume of a cone.

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Use the slicing method to derive the formula for the volume of a tetrahedron with side length a .

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Use the disk method to derive the formula for the volume of a trapezoidal cylinder.

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Explain when you would use the disk method versus the washer method. When are they interchangeable?

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For the following exercises, draw a typical slice and find the volume using the slicing method for the given volume.

A pyramid with height 6 units and square base of side 2 units, as pictured here.

This figure is a pyramid with base width of 2 and height of 6 units.

8 units 3

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A pyramid with height 4 units and a rectangular base with length 2 units and width 3 units, as pictured here.

This figure is a pyramid with base width of 2, length of 3, and height of 4 units.
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Questions & Answers

determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
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Abdul
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
find the values of c such that the graph of f(x)=x^4+2x^3+cx^2+2x+2
Ramya Reply
anyone to explain some basic in calculus
Adegoke Reply
I can
Debdoot
A conical container of radius 10 ft and height 30 ft is filled with water to a depth of 15 ft. How much work is required to pump all the water out through a hole in the top of the container if the unit weight of the water is 62.4 lb/ft^3?
Milca Reply
hi am new here I really wants to know how the solve calculus
IBRAHIM
me too. I want to know calculation involved in calculus.
Katiba
evaluate triple integral xyz dx dy dz where the domain v is bounded by the plane x+y+z=a and the co-ordinate planes
BAGAM Reply
So how can this question be solved
Eddy
i m not sure but it could be xyz/2
Leo
someone should explain with a photo shot of the working pls
Adegoke
I think we should sort it out.
Eunice
Eunice Toe you can try it if you have the idea
Adegoke
how
Eunice
a^6÷8
Muzamil
i think a^6 ÷ 8
Muzamil
Practice Key Terms 5

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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