Solid Shape Volume by Disk Method Problem

In summary, the problem involves finding the volume of a solid with an equilateral triangle base and square cross sections intersecting perpendicular to the x-axis. The correct solution involves finding the area of the cross section and integrating it from the correct limits to find the volume.
  • #1
harrietstowe
46
0

Homework Statement


The base of a certain solid is an equilateral triangle of side a, with one vertex at the origin and an altitude along the x-axis. Each plane perpendicular to the x-axis intersects the solid in a square cross section with one side in the base of the solid. Find the volume.


Homework Equations





The Attempt at a Solution



I am pretty unsure about this problem but what I did was:
Let a be one side of this equilateral triangle
Let f(x) = (c*x)/(a*sqrt(3))
Let g(x) = (-c*x)/(a*sqrt(3))
so a*sqrt(3) is the altitude and c is a constant
Solve for c:
tan(60 deg)= (a*sqrt(3))/c
c = a
Sub that into f(x) and g(x)
f(x) = (sqrt(3)*x)/3
g(x) = (-sqrt(3)*x)/3
The next thing I did was to subtract the two functions to get the area
A=f(x)-g(x)= (2*sqrt(3)*x)/3
the lower limit of integration will be 0
the upper limit of integration will be a
So I took the integral (with respect to x) from 0 to a of the Area squared and I got (4a^3)/9
The correct answer though is (sqrt(3)*a^3)/6
I hope my set up wasn't completely off but I guess you guys can give me so feedback.
Thanks
 
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  • #2
The integration limit is not correct. If you made a drawing and saw the picture of the body you would have found out the correct limits of integration.

ehild
 

Related to Solid Shape Volume by Disk Method Problem

1. What is the disk method for finding volume of a solid shape?

The disk method is a mathematical technique for finding the volume of a solid shape by using disks to approximate the shape. It involves dividing the shape into infinitesimally thin slices (disks) and adding up the volumes of these disks to find the total volume of the shape. This method is commonly used for finding the volume of solids of revolution, such as cylinders and cones.

2. How do you set up a disk method problem?

To set up a disk method problem, you first need to identify the axis of rotation. This is the line around which the shape will be rotated to form a solid of revolution. Next, you will need to determine the limits of integration, which are the starting and ending points for the integration process. These are typically the x or y values that define the boundaries of the shape. Finally, you will need to determine the function that represents the cross-sectional area of the shape, which will be used to calculate the volume of each disk.

3. What is the formula for calculating the volume of a disk?

The formula for calculating the volume of a disk is V = πr²h, where π is the mathematical constant pi, r is the radius of the disk, and h is the height of the disk. This formula can be used to calculate the volume of each disk in a disk method problem, which are then added together to find the total volume of the solid shape.

4. When should the disk method be used to find volume?

The disk method is most commonly used to find the volume of solids of revolution, such as cylinders and cones. It should be used when the shape can be rotated around an axis of rotation to form a solid of revolution. This method is also helpful when the function that represents the cross-sectional area of the shape is difficult to integrate using other methods, such as the shell method.

5. What are some common mistakes to avoid in a disk method problem?

One common mistake to avoid in a disk method problem is not correctly setting up the limits of integration. It is important to carefully identify the boundaries of the shape and make sure they are correctly incorporated into the integral. Another mistake to avoid is not using the correct formula for the volume of a disk, which can result in an incorrect final answer. Finally, it is important to check the units of measurement and ensure that they are consistent throughout the problem.

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