# Revolving regions horizontally and vertically

• Blonde1551
In summary, the conversation discusses using the disk and washer methods to find the volume of a region bounded by the graphs y=√X, y=0, and x=3 when rotated around x=3 and x=6. The correct radius values for both cases are discussed, as well as the incorrect answers the speaker obtained and their search for help in finding their mistake.
Blonde1551

## Homework Statement

The question asks me to revolve the the region bounded by the graphs y=√X y=0 and x=3 around a) the line x=3 and then b) around the line x=6

## Homework Equations

I know I am supposed to use the disk method or the washer method for this, which is ∏ ∫ R(y)^2-r(y)^2 from y=c to y=d.

## The Attempt at a Solution

for a, I set up R(y)=y^2. I set up the integral from y=0 to y= √3 ∏ ∫(y^2)^2
but when I plugged in the values I got 9.79 units cubed. In the back of my calculus book, however, it gives a different answer. I can't figure out what I did wrong.

For b, I set R(y)=6 and r(y)=(6-y^2). I set up the the integral from y=0 to y=√3
∏∫6^2-((6-y^2)^2)

I got an answer of 55.5 units cubed. Again, this was not the correct answer given to me in the back of my book.I can't figure out what I did wrong. Any help would be greatly appreciated.

For a, you don't quite have the correct value for the radius. R(y) = y2 is correct for the function, but remember that if you're revolving around x = 3, then the radius is going to be the distance between R(y) and 3.

For b, I think it's the same type of thing. The outer radius should be the distance between x = 6 and R(y), and the inner radius should be the distance between x = 6 and x = 3 (which is the inner boundary).

## 1. What is a revolving region?

A revolving region is a two-dimensional shape that rotates around a fixed point, creating a three-dimensional solid or object. This process is also known as rotation or revolution.

## 2. How do you define a revolving region horizontally?

A revolving region is defined horizontally when the axis of rotation is parallel to the x-axis. This results in a solid that is created by rotating a two-dimensional shape around a horizontal line.

## 3. What does it mean to revolve a region vertically?

Revolving a region vertically means that the axis of rotation is perpendicular to the x-axis, causing the shape to rotate around a vertical line. This results in a solid that is created by rotating a two-dimensional shape around a vertical line.

## 4. What is the difference between horizontal and vertical revolving regions?

The main difference between horizontal and vertical revolving regions is the axis of rotation. In a horizontal revolving region, the axis is parallel to the x-axis, while in a vertical revolving region, the axis is perpendicular to the x-axis. This results in different shapes being created.

## 5. How are revolving regions used in real life?

Revolving regions have many real-life applications, such as in manufacturing and engineering. For example, a lathe machine uses revolving regions to shape and cut metal or wood into various objects. They are also used in architecture and design to create 3D models of buildings and other structures.

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