Density & Integration.... Help?

In summary, the conversation discusses finding the total mass of sawdust in a conical hole with a depth of 19 meters and radius of 16 meters. The density of the sawdust varies with depth according to the formula ρ(x) = 2.1 + 1.2e^(-1.2x) kg/m^3. The solution involves setting up and solving an integral using the relationship between mass, density, and volume.
  • #1
sunnnystrong
54
6

Homework Statement



A hole in the ground in the shape of an inverted cone is 19 meters deep and has radius at the top of 16 meters. The cone is filled to the top with sawdust. The density of the sawdust depends upon the depth, x, following the formula ρ(x) = 2.1 + 1.2e^(-1.2x) kg/m^3. Find the total mass of sawdust in the conical hole.

Homework Equations



mass = density * volume

The Attempt at a Solution


[/B]
So I'm just confused as to how to set up my integral?

I want to find the def. ∫ density*volume from 0 to 9
 

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  • #2
Using the relationship I posted above ^^^

radius of cone = 16m
height of cone = 19m
s = (16/19)(19-x)
So my integral would be...
 

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  • #3
Anyone? D:
 
  • #4
*** Figured it out. all you had to do was square the constant (16/19) as well.
 

Related to Density & Integration.... Help?

1. What is density?

Density is a measure of how much mass is contained in a given volume. It is calculated by dividing the mass of an object by its volume. The units for density are typically grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3).

2. How is density related to integration?

Density plays a crucial role in the process of integration, as it is used to determine the total mass of an object or system. When integrating, the density of each element or component is multiplied by its volume and then summed together to find the total mass.

3. What are some real-life applications of density and integration?

Density and integration are used in a variety of fields, such as engineering, physics, and chemistry. Some examples of real-life applications include determining the density of a material for construction purposes, calculating the mass of a compound in chemistry experiments, and finding the center of mass of an object for designing structures.

4. How does density affect the buoyancy of an object?

Density is a key factor in determining the buoyancy of an object. If the density of an object is less than the density of the liquid it is placed in, it will float. If the density of the object is greater than the density of the liquid, it will sink. This is due to the fact that the buoyant force acting on an object is equal to the weight of the fluid it displaces.

5. Can density and integration be used to find the volume of irregularly shaped objects?

Yes, density and integration can be used to find the volume of irregularly shaped objects. This can be done by dividing the object into smaller, regular shapes for which the volume can be easily calculated. The volumes of these smaller shapes are then integrated to find the total volume of the irregular object.

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