Volume of a function around an axis

In summary, when finding the volume and describing the shape of the object formed by the function f(x) = ½ x²+2 when rotated around the x=3 axis and bounded by the region between x=0 and x=4, the correct formula to use is 2π ∫Pxhx dx between x=0 and x=4. The function should be set in terms of y due to its rotation around the vertical axis, and the radius is determined by subtracting 3 from the function. The volume can be calculated using the formula π ∫ sqrt(2x-2) - 6sqrt(2x-2) - 13 dy from y=0 to y=4, resulting in a
  • #1
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Given this problem:

Find the volume and describe the shape of the object formed by the function f(x) = ½ x²+2 when the function is rotated around the x =3 axis and bounded by th region between the x = 0 and x = 4.

I am not sure if I am thihnking about this correctly. I know in order to find the volume, I must use the formula 2π ∫Pxhx dx between x = 0 and x = 4. However, since it is rotatated around the x= 3 axis, do I use that as my starting point. Meaning, do I make 3 equal to zero and make the x boundaries as the difference from 3 to their points?
 
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  • #2
The axis of rotation is inside the region to be rotated so that the solid would intersect itself, did you copy the problem down right?
 
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  • #3
Sorry! Never mind the question. I was looking at the problem wrong. The function is bounded by y = 0 and y = 4, not x. Now it is a piece of cake!
 
  • #4
I'm back and I've carefully checked to make sure I am reading it correctly.

First I set function in terms of y because of its rotation around the verticle axis, which gives me
y = sqrt(2x-2) Then I subtracted 3 because it revolves around x = 3. This is my radius, so I square it to get my area ( and multiply by Pi)

which gives [sqrt(2x-2 - 3]^ 2 = 2x - 6sqrt(2x-2) - 13

So my volume is π ∫ sqrt(2x-2) - 6sqrt(2x-2) - 13 dy from y = 0 to y = 4.

I get [ x² -13x - 4sqrt(2x-2)^(3/2)

Since the lowest point of this function is 2 on ther y axis, the volume should be the same whether I use the bounds of 2 to 4, or 0 to 4, but I keep gwetting differeent results for those two limits.
 

What is the definition of the "volume of a function around an axis"?

The volume of a function around an axis is the measure of space occupied by the solid formed when a two-dimensional shape is rotated around a line, or axis, in three-dimensional space.

How is the volume of a function around an axis calculated?

The volume of a function around an axis can be calculated using the disk or washer method. The disk method involves slicing the solid into infinitesimally thin disks perpendicular to the axis, finding the volume of each disk, and then integrating to find the total volume. The washer method involves slicing the solid into infinitesimally thin washers (or annuli) parallel to the axis, finding the volume of each washer, and then integrating to find the total volume.

What is the difference between the disk and washer method for calculating volume of a function around an axis?

The main difference between the disk and washer method is in the shape of the infinitesimally thin slices used to calculate volume. In the disk method, the slices are disks perpendicular to the axis, while in the washer method, the slices are washers (or annuli) parallel to the axis. This can result in different integrals being used to calculate the total volume, although both methods will yield the same answer.

Can the volume of a function around an axis be negative?

No, the volume of a function around an axis cannot be negative. By definition, volume is a measure of space and cannot have a negative value. If the calculated volume is negative, it may indicate an error in the calculation.

What are some real-life applications of calculating volume of a function around an axis?

The concept of volume of a function around an axis is used in various fields such as engineering, architecture, and physics. For example, it is used in the design of structures like bridges and tunnels, in determining the displacement of water in a container, and in calculating the moment of inertia of a rotating object. It is also used in computer graphics to create and animate 3D objects.

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