# Finding the area of revolution

• farmd684
In summary, the conversation discusses a problem involving finding the volume of a solid of revolution obtained by rotating a region enclosed by y = sinx, y = cosx, and the x-axis around the y-axis and x-axis. The region is not clearly described, causing confusion for the person asking for help. The expert suggests getting clarification from the instructor or making an assumption and using techniques such as concentric shells or disks to calculate the volume.
farmd684
How can i get the volume of this problem

## Homework Statement

y = cosx, y = sinx , x = pie/2 , and the y-axis finding the volume about x & y axis

If you want help, you need to show what you have tried to do.

Also, your problem description suggests that maybe you are trying to find two different volumes of revolution (and not the area of revolution, as is mistakenly stated in your title).

sorry for the mistake. It should be volume revolving about x and y-axis to be found separately.

I have done graphing all of the equations in the coordinate axis it gives me a square like ( y = sinx , y = cosx upto pi/2 and x= pi/2 and the y-axis )shape. Then which of the function i will use to find out my volume ?

Last edited:
You can start by giving us the exact problem description as stated in your book or wherever you got the problem.

The region that is going to be revolved around each axis (separately) is not a square.

BTW, this Greek letter, $\pi$, is pi, not pie.

The exact problem is :

Find the region enclosed by y = sinx, y = cosx , x= pi/2 and y axis.

Hence find the the volume of the solid of revolution obtained by rotating the region

OK, the description still seems vague to me, so I believe the region is bounded by the graphs of y = sinx, y = cosx, and the x-axis, between x = 0 and x = pi/2.

This region has a sort of triangular shape but with two curved sides, with vertices at (0, 0), (pi/4, sqrt(2)/2, and (pi/2, 0).

If you revolve this region around the y-axis, you get a sort of donut-shaped solid. There are two ways to get the typical volume element: concentric shells and disks.

By concentric shells, the typical volume element is $\Delta$V = pi*x^2*f(x)*$\Delta$x. f(x) represents the height of the shell, which is sin(x) on the first part of the interval and cos(x) on the second part of the interval. This means you will need to use two separate integrals, one for each subinterval, to represent the volume.

By disks, the typical volume element is $\Delta$V = pi*(R^2 - r^2)$\Delta$y. R represents the radius (the x-value) from the y-axis to curve that is farther away, and r represents the radius (the x-value) from the y-axis to curve that is nearer. Since the thickness of the disk is $\Delta$y, the two x values for the radii need to be written in terms of y, so you'll need to use inverse functions. IOW since y = cos(x) for the more distant curve, x = cos-1(y) gives the radius R. It's going to be much more complicated to use disks, so I would recommend using shells for this part of the problem.

To find the volume when the region is rotated around the x-axis, use similar reasoning to determine the typical volume element, which can be gotten using the same two techniques I already used.

I m sure that the region is enclosed by y = sinx, y = cosx , x= pi/2 and y axis.

Thanks

farmd684 said:
I m sure that the region is enclosed by y = sinx, y = cosx , x= pi/2 and y axis.
The y-axis is x = 0.
farmd684 said:
Thanks
IMO this problem is very poorly worded, because it does not describe the region to be rotated in an unambiguous fashion. Your choices are to:
1. Get clarification from your instructor.
2. Make an assumption (and state it explicitly) about what you believe the problem is asking, and continue on from there.

Mark44 said:
The y-axis is x = 0.

IMO this problem is very poorly worded, because it does not describe the region to be rotated in an unambiguous fashion. Your choices are to:
1. Get clarification from your instructor.
2. Make an assumption (and state it explicitly) about what you believe the problem is asking, and continue on from there.

What clarification do i need from my instructor ? This is the given problem and i m asked to solve it.

I just want to know if it is possible or not to get the volume of the given region if it is then how can i do it if it is not then i have to submit report to instructor that it is impossible to get this volume.

farmd684 said:
What clarification do i need from my instructor ? This is the given problem and i m asked to solve it.
Clarification on what exactly is the region to be revolved. As I said before, the description is poorly worded.

If you want to proceed without getting clarification from you instructor, you can make an assumption about what you think the problem is saying, and then continue from there. I have laid out the steps for you. Have you read them? I'm somewhat surprised that this is post #10 in this thread and it doesn't seem that you have done anything in the way of working on this problem.

farmd684 said:
I just want to know if it is possible or not to get the volume of the given region if it is then how can i do it if it is not then i have to submit report to instructor that it is impossible to get this volume.
Which given region? The problem description is unclear about the region to be revolved. That's what you should get clarification on. Once you know exactly what the region is, then you can find each volume of revolution using what I said in a previous post.

Mark44 said:
Clarification on what exactly is the region to be revolved. As I said before, the description is poorly worded.

If you want to proceed without getting clarification from you instructor, you can make an assumption about what you think the problem is saying, and then continue from there. I have laid out the steps for you. Have you read them? I'm somewhat surprised that this is post #10 in this thread and it doesn't seem that you have done anything in the way of working on this problem.

Which given region? The problem description is unclear about the region to be revolved. That's what you should get clarification on. Once you know exactly what the region is, then you can find each volume of revolution using what I said in a previous post.

I have already done the problem several times and i m not sure if i m right or wrong. This is a homework like i have to submit it.

Y=sinx , y = cosx , x= pi/2 and the y-axis is the given region. This is giving me a square like shape ? i m confused how to break the volume of the region because two sin and cos curves are intersecting the the given region and also i have to consider the y axis.

Please let me be clear how can i will break up the integral & the solution you gave is your made problem not mine & it is done with x-axis. Thats y is thread is getting bigger. Because i haven't found any solution yet.

farmd684 said:
I have already done the problem several times and i m not sure if i m right or wrong. This is a homework like i have to submit it.
Well, I certainly can't tell if you're right or wrong because I haven't seen even the tiniest amount of your work.
farmd684 said:
Y=sinx , y = cosx , x= pi/2 and the y-axis is the given region.
What I have repeatedly said is that the description you gave does not define an unambiguous region. There are three separate regions: one is below y = cos(x) and above y = sin(x) for x between 0 and pi/4, another is below y = sin(x) and above y = cos(x) for x between pi/4 and pi/2, and a third region that lies between the first two regions. Is the region we're suppose to revolve the first two and not the third or is it just the third, or what?
farmd684 said:
This is giving me a square like shape ?
What square like shape? What are you talking about? I don't see anything that looks remotely like a square.
farmd684 said:
i m confused how to break the volume of the region because two sin and cos curves are intersecting the the given region and also i have to consider the y axis.
Why are you talking about breaking the volume of the region? The region - whatever it is - is two-dimensional, so has area but not volume. When we figure out what the region is, then we can revolve it around the y-axis (first problem) and get an integral that represents the volume. Same thing for the second problem, which entails revolving the region around the x-axis.
farmd684 said:
Please let me be clear how can i will break up the integral & the solution you gave is your made problem not mine & it is done with x-axis. Thats y is thread is getting bigger. Because i haven't found any solution yet.

This is not my made-up problem - I'm trying to understand what your problem is asking for. I have also said that when you have a problem you don't understand, you can either ask the instructor for clarification (have you done that?) or make some assumptions about what you think the problem is asking and go from there (have you done that?).

Mark44 said:
What I have repeatedly said is that the description you gave does not define an unambiguous region. There are three separate regions: one is below y = cos(x) and above y = sin(x) for x between 0 and pi/4, another is below y = sin(x) and above y = cos(x) for x between pi/4 and pi/2, and a third region that lies between the first two regions. Is the region we're suppose to revolve the first two and not the third or is it just the third, or what?

These are the three regions i got. I guess i have to revolve them all.

Sorry it is not a square like shape.

## 1. What is "Finding the Area of Revolution"?

"Finding the Area of Revolution" is a mathematical concept used to determine the surface area of a three-dimensional shape that has been created by rotating a two-dimensional shape around an axis.

## 2. How is the area of revolution calculated?

The area of revolution is calculated by using an integral to find the sum of infinitely thin slices of the three-dimensional shape. The formula for finding the area of revolution varies depending on the shape being rotated and the axis of rotation.

## 3. What are some common shapes that require finding the area of revolution?

Some common shapes that require finding the area of revolution include cylinders, cones, spheres, and toroids. These shapes can be created by rotating a circle, rectangle, or other two-dimensional shape around a central axis.

## 4. Why is finding the area of revolution important?

Finding the area of revolution is important in many fields, including engineering, physics, and architecture. It allows for the accurate calculation of surface area, which is essential in designing and constructing various objects and structures.

## 5. What are some real-life examples of finding the area of revolution?

Real-life examples of finding the area of revolution include calculating the surface area of a water tank, determining the amount of material needed to create a cylindrical pipe, and estimating the surface area of a rollercoaster loop. It can also be used in calculating the volume of certain shapes, such as a sphere or cone.

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