Volumes of Solids with Known Cross Section Project

In summary, the problem involves finding the volume of a solid given by the cross sections of isosceles right triangles with one vertex at y= √x, boundaries of x=0 and x=9, and a thickness of dx. The volume can be found by integrating the area of each cross section, which is (x/2)dx.
  • #1
enn
1
0
I'm trying to get started on this project but am totally confused about how to find the volume of the solid. All the information I was given was the following:

y= √x

boundaries: 0,9

cross sections: isosceles right triangle

how the hell do I get started?!
 
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  • #2
enn said:
I'm trying to get started on this project but am totally confused about how to find the volume of the solid. All the information I was given was the following:

y= √x

boundaries: 0,9

cross sections: isosceles right triangle

how the hell do I get started?!

You use that the integral of area is equal to volume. You certainly need a little more information. My guess is that you are supposed to assume one of the sides of the triangle is between (x,0) and (x,sqrt(x)). Whether it's the hypotenuse of the triangle or the side is yours to guess unless they gave you a little more info.
 
  • #3
enn said:
I'm trying to get started on this project but am totally confused about how to find the volume of the solid. All the information I was given was the following:

y= √x

boundaries: 0,9

cross sections: isosceles right triangle

how the hell do I get started?!
You are doing just about everything wrong here. First, you are being rude- not a good way to ask for help. Second, I don't believe this was "all the information" you were given! For example, I'll bet you were told what "y= √x" means and you don't tell us that. I suspect you were told that the right angle of that "iososceles right triangle" lies on the the x-axis and another vertex on the graph of y= √x. Also, I'll bet that you were NOT told "boundaries: 0, 9" but were told that one end of the solid is at x= 0 and the other at x= 9.

If that is true then an isosceles right triangle with right angle on the x-axis and another vertex at y= √x has both legs of length √x and so area (1/2)bh= (1/2)(√x)(√x)= x/2. If we imagine one cross section "slab" as having thickness "dx" then its volume is (x/2)dx. Find the whole volume by integrating that.
 

1. What is the purpose of a "Volumes of Solids with Known Cross Section Project"?

The purpose of this project is to calculate the volume of a three-dimensional object using known cross sections. This can be used in various fields of science, such as engineering and architecture, to determine the volume of complex shapes.

2. How do you determine the volume of a solid with known cross sections?

To determine the volume of a solid with known cross sections, you first need to identify the shape of the cross section. Then, you can use the appropriate formula for that shape (e.g. area of a circle for a circular cross section) to calculate the volume of each cross section. Finally, you can add up all the volumes of the cross sections to get the total volume of the solid.

3. What types of cross sections can be used in this project?

Any two-dimensional shape can be used as a cross section in this project, as long as it is perpendicular to the axis of rotation. Commonly used shapes include circles, triangles, rectangles, and semi-circles.

4. Can this project be used for irregularly shaped objects?

Yes, this project can be used for irregularly shaped objects as long as the cross sections are known and perpendicular to the axis of rotation. However, the calculations may be more complex and require more advanced mathematical methods.

5. What are some real-life applications of this project?

This project has many real-life applications, such as calculating the volume of a water tank, determining the capacity of a container, or finding the volume of a curved pipe. It is also used in various fields of science and engineering, such as architecture, construction, and manufacturing.

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