# How Do You Calculate the Volume of a Solid Revolved Around Different Axes?

• jaredm2012
In summary, the volume of the solid generated by revolving the region bounded by y=x^1/2 and the lines y=2 and x=0 about the line x=4 is 0.
jaredm2012
Nevermind, I finally figured it out!

## Homework Statement

Find the volume of the solid generated by revolving the region bounded by y=x^1/2 and the lines y=2 and x=0 about:

a. the x-axis
b. the y-axis
c. the line y=2
d. the line x=4

## Homework Equations

integral from a--b pi[R(x)]^2dx

integral from a--b pi([R(x)]^2 - [r(x)]^2)dx

## The Attempt at a Solution

I solved a, b, and c just fine.

a = 8pi
b = 32pi/5
c = 8pi/3

However I am confused about part d, particularly how to find the radius. The book gives the answer as 224pi/15, but I get 256pi/15.

I did the following:

pi*integral from 0 to 2 (4-y^2)^2dy, which gave me 256pi/15 when I worked it out. I may just be messing up the math, though I did the problem a few times the same way and got the same answer every time. I also tried using pi*integral from 0 to 2 16-y^4, but this also yielded the wrong answer. Basically I am not sure how to find the radius of the region it is asking for, when rotating around the line x=4.

Last edited:
Can someone explain how to do this?Answer:In order to find the radius of the region when rotating around the line x=4, you need to calculate the distance from the line x=4 to the boundary of the region. This distance is equal to 4-y^2, where y is the upper boundary of the region (in this case, y=2). Therefore, the radius of the region when rotating around x=4 is 4-2^2 = 0. Therefore, the volume of the solid generated by revolving the region bounded by y=x^1/2 and the lines y=2 and x=0 about the line x=4 is 0, since the radius of the region is 0.

## 1. What is the disc/washer method problem?

The disc/washer method problem is a calculus problem that involves finding the volume of a solid of revolution by using either discs or washers to approximate the shape of the solid.

## 2. When is the disc/washer method used?

The disc/washer method is typically used when the solid being revolved has a known cross-sectional area and a known axis of revolution, making it easier to calculate the volume.

## 3. What is the difference between the disc method and the washer method?

The disc method is used when the cross-sectional area is a circle, while the washer method is used when the cross-sectional area is a ring or annulus. This means that the disc method is used when the axis of revolution is on the edge of the solid, while the washer method is used when the axis of revolution is within the solid.

## 4. How is the disc/washer method problem solved?

The disc/washer method problem is solved by integrating the cross-sectional area function over the interval of revolution and using the limits of integration to find the bounds of the solid. The resulting integral is then evaluated to find the volume of the solid.

## 5. What are some common mistakes when using the disc/washer method?

Some common mistakes when using the disc/washer method include forgetting to square the radius when calculating the cross-sectional area, using the wrong axis of revolution, and not setting up the integral correctly. It is important to carefully consider the shape of the solid and the limits of integration before attempting to solve the problem.

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