Engineering Mathematics Exam > Engineering Mathematics Videos > Washer method rotating around non-axis - Mathematics
Washer method rotating around non-axis - Video Lecture - Engineering
FAQs on Washer method rotating around non-axis - Mathematics
| 1. What is the washer method in mathematics engineering? |  |
Ans. The washer method is a technique used in mathematics engineering to find the volume of a solid of revolution by rotating a region between two curves around a non-axis line. It involves slicing the solid into thin washers, calculating the volume of each washer, and then adding up all the volumes to find the total volume.
| 2. How does the washer method differ from the disk method? | |
Ans. The washer method and the disk method are both used to find volumes of solids of revolution. The main difference lies in the shape of the cross-sections. The disk method involves using circular disks to approximate the cross-sectional area, while the washer method uses washers (or annuli) to approximate the cross-sectional area. The washers have a hole in the middle, which accounts for the region between the two curves.
| 3. Can the washer method be applied to any shape? | |
Ans. No, the washer method can only be applied to 2D shapes that can be rotated around a non-axis line to form a solid of revolution. The shapes must be bounded by two curves and the region between these curves is revolved to create the solid. Examples of shapes that can be used with the washer method include circles, ellipses, and parabolas.
| 4. How do you set up the integral for the washer method? | |
Ans. To set up the integral for the washer method, you need to express the equations of the two curves that bound the region of rotation. Then, you integrate the function that represents the difference between the outer and inner curves squared (r^2 - R^2) over the interval of rotation. The integral is typically set up with respect to the variable that determines the axis of rotation.
| 5. What are the limitations of the washer method? | |
Ans. The washer method has some limitations. It is primarily used for finding volumes of solids of revolution and is not applicable to other types of problems. Additionally, the washer method can be more complex to set up compared to the disk method, especially when dealing with curved or asymmetrical shapes. It requires a good understanding of calculus and geometry to accurately apply the washer method in engineering mathematics.
Text Transcript from Video
Now let's do a really
interesting problem.
So I have y equals x,
and y is equal to x
squared minus 2x
right over here.
And we're going to
rotate the region
in between these two functions.
So that's this region
right over here.
And we're not going to rotate
it just around the x-axis,
we're going to rotate it around
the horizontal line y equals 4.
So we're going to
rotate it around this.
And if we do that, we'll get
a shape that looks like this.
I drew it ahead of time, just
so I could draw it nicely.
And as you can see, it looks
like some type of a vase
with a hole at the bottom.
And so what we're going to do
is attempt to do this using,
I guess you'd call it
the washer method which
is a variant of the disk method.
So let's construct a washer.
So let's look at a given x.
So let's say an x
right over here.
So let's say that we're
at an x right over there.
And what we're
going to do is we're
going to rotate this region.
We're going to give
it some depth, dx.
So that is dx.
We're going to rotate
this around the line y
is equal to 4.
So if you were to visualize it
over here, you have some depth.
And when you rotate it
around, the inner radius
is going to look like the
inner radius of our washer.
It's going to look
something like that.
And then the outer
radius of our washer
is going to contour
around x squared minus 2x.
So it's going to
look something--
my best attempt
to draw it-- it's
going to look
something like that.
And of course, our washer
is going to have some depth.
So let me draw the depth.
So it's going to
have some depth, dx.
So this is my best attempt at
drawing some of that the depth.
So this is the
depth of our washer.
And then just to make the face
of the washer a little bit
clearer, let me do it
in this green color.
So the face of the
washer is going
to be all of this business.
All of this business is going
to be the face of our washer.
So if we can figure
out the volume of one
of these washers for a
given x, then we just
have to sum up all of
the washers for all
of the x's in our interval.
So let's see if we can
set up the integral,
and maybe in the
next video we'll
just forge ahead and actually
evaluate the integral.
So let's think about the
volume of the washer.
To think about the
volume of the washer,
we really just have to
think about the area
of the face of the washer.
So area of "face"--
put face in quotes--
is going to be equal to what?
Well, it would be the
area of the washer--
if it wasn't a washer,
if it was just a coin--
and then subtract out
the area of the part
that you're cutting out.
So the area of the
washer if it didn't
have a hole in the
middle would just
be pi times the
outer radius squared.
It would be pi times
this radius squared,
that we could call
the outer radius.
And since it's a washer,
we need to subtract out
the area of this inner circle.
So minus pi times
inner radius squared.
So we really just
have to figure out
what the outer and inner radius,
or radii I should say, are.
So let's think about it.
So our outer radius is
going to be equal to what?
Well, we can visualize
it over here.
This is our outer
radius, which is also
going to be equal to
that right over there.
So that's the distance
between y equals
4 and the function that's
defining our outside.
So this is essentially,
this height right over here,
is going to be equal to 4
minus x squared minus 2x.
I'm just finding the distance
or the height between these two
functions.
So the outer radius is
going to be 4 minus this,
minus x squared minus
2x, which is just 4
minus x squared plus 2x.
Now, what is the inner radius?
What is that going to be?
Well, that's just going to
be this distance between y
equals 4 and y equals x.
So it's just going
to be 4 minus x.
So if we wanted to find
the area of the face of one
of these washers for a
given x, it's going to be--
and we can factor
out this pi-- it's
going to be pi times the
outer radius squared,
which is all of this
business squared.
So it's going to be 4 minus
x squared plus 2x squared
minus pi times
the inner radius--
although we factored
out the pi-- so
minus the inner radius squared.
So minus 4 minus x squared.
So this will give us
the area of the surface
or the face of one
of these washers.
If we want the volume of
one of those watchers,
we then just have to
multiply times the depth, dx.
And then if we want to actually
find the volume of this entire
figure, then we just have to
sum up all of these washers
for each of our x's.
So let's do that.
So we're going to sum
up the washers for each
of our x's and take the
limit as they approach zero,
but we have to make sure
we got our interval right.
So what are these-- we care
about the entire region
between the points
where they intersect.
So let's make sure
we get our interval.
So to figure out our
interval, we just
say when does y equal
x intersect y equal
x squared minus 2x?
Let me do this in
a different color.
We just have to
think about when does
x equal x squared minus 2x.
When are our two functions
equal to each other?
Which is equivalent
to-- if we just
subtract x from
both sides, we get
when does x squared
minus 3x equal 0.
We can factor out an x
on the right hand side.
So this is going to be when does
x times x minus 3 equal zero.
Well, if the product is equal
to 0, at least one of these
need to be equal to 0.
So x could be equal to 0,
or x minus 3 is equal to 0.
So x is equal to 0
or x is equal to 3.
So this is x is 0, and
this right over here
is x is equal to 3.
So that gives us our interval.
We're going to go
from x equals 0
to x equals 3 to get our volume.
In the next video,
we'll actually
evaluate this integral.
interesting problem.
So I have y equals x,
and y is equal to x
squared minus 2x
right over here.
And we're going to
rotate the region
in between these two functions.
So that's this region
right over here.
And we're not going to rotate
it just around the x-axis,
we're going to rotate it around
the horizontal line y equals 4.
So we're going to
rotate it around this.
And if we do that, we'll get
a shape that looks like this.
I drew it ahead of time, just
so I could draw it nicely.
And as you can see, it looks
like some type of a vase
with a hole at the bottom.
And so what we're going to do
is attempt to do this using,
I guess you'd call it
the washer method which
is a variant of the disk method.
So let's construct a washer.
So let's look at a given x.
So let's say an x
right over here.
So let's say that we're
at an x right over there.
And what we're
going to do is we're
going to rotate this region.
We're going to give
it some depth, dx.
So that is dx.
We're going to rotate
this around the line y
is equal to 4.
So if you were to visualize it
over here, you have some depth.
And when you rotate it
around, the inner radius
is going to look like the
inner radius of our washer.
It's going to look
something like that.
And then the outer
radius of our washer
is going to contour
around x squared minus 2x.
So it's going to
look something--
my best attempt
to draw it-- it's
going to look
something like that.
And of course, our washer
is going to have some depth.
So let me draw the depth.
So it's going to
have some depth, dx.
So this is my best attempt at
drawing some of that the depth.
So this is the
depth of our washer.
And then just to make the face
of the washer a little bit
clearer, let me do it
in this green color.
So the face of the
washer is going
to be all of this business.
All of this business is going
to be the face of our washer.
So if we can figure
out the volume of one
of these washers for a
given x, then we just
have to sum up all of
the washers for all
of the x's in our interval.
So let's see if we can
set up the integral,
and maybe in the
next video we'll
just forge ahead and actually
evaluate the integral.
So let's think about the
volume of the washer.
To think about the
volume of the washer,
we really just have to
think about the area
of the face of the washer.
So area of "face"--
put face in quotes--
is going to be equal to what?
Well, it would be the
area of the washer--
if it wasn't a washer,
if it was just a coin--
and then subtract out
the area of the part
that you're cutting out.
So the area of the
washer if it didn't
have a hole in the
middle would just
be pi times the
outer radius squared.
It would be pi times
this radius squared,
that we could call
the outer radius.
And since it's a washer,
we need to subtract out
the area of this inner circle.
So minus pi times
inner radius squared.
So we really just
have to figure out
what the outer and inner radius,
or radii I should say, are.
So let's think about it.
So our outer radius is
going to be equal to what?
Well, we can visualize
it over here.
This is our outer
radius, which is also
going to be equal to
that right over there.
So that's the distance
between y equals
4 and the function that's
defining our outside.
So this is essentially,
this height right over here,
is going to be equal to 4
minus x squared minus 2x.
I'm just finding the distance
or the height between these two
functions.
So the outer radius is
going to be 4 minus this,
minus x squared minus
2x, which is just 4
minus x squared plus 2x.
Now, what is the inner radius?
What is that going to be?
Well, that's just going to
be this distance between y
equals 4 and y equals x.
So it's just going
to be 4 minus x.
So if we wanted to find
the area of the face of one
of these washers for a
given x, it's going to be--
and we can factor
out this pi-- it's
going to be pi times the
outer radius squared,
which is all of this
business squared.
So it's going to be 4 minus
x squared plus 2x squared
minus pi times
the inner radius--
although we factored
out the pi-- so
minus the inner radius squared.
So minus 4 minus x squared.
So this will give us
the area of the surface
or the face of one
of these washers.
If we want the volume of
one of those watchers,
we then just have to
multiply times the depth, dx.
And then if we want to actually
find the volume of this entire
figure, then we just have to
sum up all of these washers
for each of our x's.
So let's do that.
So we're going to sum
up the washers for each
of our x's and take the
limit as they approach zero,
but we have to make sure
we got our interval right.
So what are these-- we care
about the entire region
between the points
where they intersect.
So let's make sure
we get our interval.
So to figure out our
interval, we just
say when does y equal
x intersect y equal
x squared minus 2x?
Let me do this in
a different color.
We just have to
think about when does
x equal x squared minus 2x.
When are our two functions
equal to each other?
Which is equivalent
to-- if we just
subtract x from
both sides, we get
when does x squared
minus 3x equal 0.
We can factor out an x
on the right hand side.
So this is going to be when does
x times x minus 3 equal zero.
Well, if the product is equal
to 0, at least one of these
need to be equal to 0.
So x could be equal to 0,
or x minus 3 is equal to 0.
So x is equal to 0
or x is equal to 3.
So this is x is 0, and
this right over here
is x is equal to 3.
So that gives us our interval.
We're going to go
from x equals 0
to x equals 3 to get our volume.
In the next video,
we'll actually
evaluate this integral.
About this Video
Apr 20, 2026 Last updated
Related Exams
Washer method rotating around non-axis - Mathematics for Engineering Mathematics 2026 is part of Engineering Mathematics preparation. The notes and questions for Washer method rotating around non-axis - Mathematics have been prepared according to the Engineering Mathematics exam syllabus. Information about Washer method rotating around non-axis - Mathematics covers all important topics for Engineering Mathematics 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Washer method rotating around non-axis - Mathematics.
Introduction of Washer method rotating around non-axis - Mathematics in English is available as part of our Engineering Mathematics preparation & Washer method rotating around non-axis - Mathematics in Hindi for Engineering Mathematics courses. Download more important topics, notes, lectures and mock test series for Engineering Mathematics Exam by signing up for free.
Description
Washer method rotating around non of covers all the important topics, helping you prepare for the Engineering Mathematics exam on EduRev. Start for free!
Information about Washer method rotating around non-axis - Mathematics
Here you can find the Washer method rotating around non-axis - Mathematics defined & explained in the simplest way possible. Besides explaining types of Washer method rotating around non-axis - Mathematics theory, EduRev gives you an ample number of questions to practice Washer method rotating around non-axis - Mathematics tests, examples and also practice Engineering Mathematics tests.
Related Searches
Washer method rotating around non-axis - Mathematics, Sample Paper, Viva Questions, Summary, Extra Questions, shortcuts and tricks, Previous Year Questions with Solutions, Semester Notes, study material, Washer method rotating around non-axis - Mathematics, ppt, Free, MCQs, pdf , video lectures, Important questions, Objective type Questions, Washer method rotating around non-axis - Mathematics, mock tests for examination, practice quizzes, Exam, past year papers;