A wire of length 8 centimeters is such that the mass between...

A wire of length 8 centimeters is such that the mass between its left end and a point $x$ centimeters to the right is $x^{3}$ grams (Figure 12). (a) What is the average density of the middle 2 -centimeter segment of this wire? Note: Average density equals mass/length. (b) What is the actual density at the point 3 centimeters from the left end?

A wire of length 8 centimeters is such that the mass between its left end and a point $x$ centimeters to the right is $x^{3}$ grams (Figure 12).
(a) What is the average density of the middle 2 -centimeter segment of this wire? Note: Average density equals mass/length.
(b) What is the actual density at the point 3 centimeters from the left end?

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Key Concepts

Continuous Mass Distribution

This concept refers to representing the mass of an object not as a constant but as a function of position along its length. In many physical applications, such as wires or rods, the mass is distributed continuously rather than being lumped at discrete points. A mass function gives the total mass accumulated from one end of the object up to a point, and it provides the basis for computing further quantities like density.

Average Density

Average density is defined as the total mass of an object (or segment of an object) divided by its length. When dealing with an interval, the average density is calculated by finding the difference in the mass function over that interval and then dividing by the length of the segment. This calculation gives a uniform density equivalent that represents the overall distribution of mass over that segment.

Point Density and Differentiation

The actual density at a specific point in a continuously distributed system is obtained by differentiating the mass function with respect to position. This derivative gives the instantaneous rate of change of mass per unit length at a given point, effectively representing the local density. Differentiation is a fundamental tool in calculus that connects cumulative quantities to their local behavior.

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