# Volume of Revolution: R around Y-Axis

• cmab
In summary, R is a region bounded by the curves y=(x-1)^2 and y=2(x-1) and revolved around the y-axis. To find the volume using the washer method, the integral must be with respect to y and the inner and outer radius must be expressed as functions of y by solving the equations for x. The correct answer is 16pi/3, but the answer sheet may be incorrect as it gives 64pi/12 (unreduced).
cmab
R is bounded by the curves y=(x-1)^2 and y=2(x-1). Axis of revolution: y-axis.

How am i supposed to do this. I know how to do the washer method, I know how to apply it when it is revolved arround the x-axis, but I don't know how to do it when it is the y axis. Can anybody explain to me the steps to do ?

cmab said:
R is bounded by the curves y=(x-1)^2 and y=2(x-1). Axis of revolution: y-axis.

How am i supposed to do this. I know how to do the washer method, I know how to apply it when it is revolved arround the x-axis, but I don't know how to do it when it is the y axis. Can anybody explain to me the steps to do ?

Your integral will be with respect to y, and you need to find the inner and outer radius as a function of y. That involves solving your equations for x.

OlderDan said:
Your integral will be with respect to y, and you need to find the inner and outer radius as a function of y. That involves solving your equations for x.

I did the reciprocals. so its x= sqrt(x)+1 and x=y/2 +1

I did everything and the result is 16pie/12, and in the answer sheet it says 64pie/12

cmab said:
I did the reciprocals. so its x= sqrt(x)+1 and x=y/2 +1

I did everything and the result is 16pie/12, and in the answer sheet it says 64pie/12

You mean x= sqrt(y)+1 and x=y/2 +1. Is that what you did? The answer given is correct (reduces to 16pi/3). When you take the difference between the lerger area and the smaller area of each washer, two terms cancel and two terms survive that need to be integrated.

Thx man, I gotted the answer but I was too tired to realize it

But I think the paper is wrong, cause it keeps giving me 64pi/12 (Unreduced)

## What is the definition of volume of revolution around the y-axis?

The volume of revolution around the y-axis refers to the volume of a three-dimensional figure created by rotating a two-dimensional shape around the y-axis, such as a curve or function.

## How is the volume of revolution around the y-axis calculated?

The volume of revolution around the y-axis is calculated using the formula V = ∫2πx * f(x) dx, where x is the distance from the y-axis to the curve and f(x) is the function representing the shape being rotated.

## What are some real-life applications of volume of revolution around the y-axis?

Volume of revolution around the y-axis can be used to calculate the volume of objects such as vases, bottles, and other cylindrical shapes. It is also used in engineering and architecture to determine the volume of structures like water towers and silos.

## What is the difference between volume of revolution around the y-axis and volume of revolution around the x-axis?

The main difference between these two types of volume of revolution is the axis of rotation. When rotating around the y-axis, the shape is revolved around a vertical line, while rotating around the x-axis involves revolving around a horizontal line. The formulas for calculation are also different.

## Are there any limitations to calculating volume of revolution around the y-axis?

One limitation is that the shape being rotated must have a continuous curve around the y-axis. Another limitation is that the formula only works for shapes that can be represented by a function. Additionally, some complex shapes may require more advanced mathematical techniques to accurately calculate the volume of revolution.

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