91Ó°ÊÓ

In physics and mathematics, a density function describes how mass per unit area is distributed over a region. For this problem, the density function is given as \( \delta(x) = 12x \), where \( x \) stands for the horizontal position of points on the thin plate.

This equation implies that the mass density at a point increases linearly with its distance from the y-axis.

Among the important properties of density functions are:

• They quantify how a substance, such as mass, is spread over a region. In this case, it tells us how the mass varies along the length of the plate.
• They are crucial for calculating various physical properties, like the center of mass and total mass, where the plate is not uniform.

Understanding the density function is essential to determining how the mass is distributed, which will later assist in calculating the center of mass accurately.

To solve problems related to area or mass bounded by curves, finding intersection points is crucial. The given problem requires intersecting a parabola \( y = x^2 \) and a line \( y = x \).

These intersection points define the limits of integration. To find where these two curves meet, solve the equation \( x^2 = x \).

• Rearrange it to \( x^2 - x = 0 \) and factor it out to get \( x(x-1) = 0 \).
• This results in two solutions: \( x = 0 \) and \( x = 1 \).

These x-values represent the interval of the region on the x-axis where both curves overlap. This forms the basis for calculating the area, and subsequently the mass and center of mass of the region. Finding intersections is a fundamental skill needed to correctly frame physical problems.

Integration is a powerful mathematical tool used to calculate quantities like area, volume, and mass, especially when dealing with variable densities.

In this problem, various integrals are set up to find the mass, and the x and y components of the center of mass. The integral for mass, \( M \), is: \[M = \int_{0}^{1} \delta(x) \Delta y \, dx\]where \( \Delta y = x - x^2 \) is the height of the strip at any position \( x \). Breaking this integral into simpler terms:

• The mass calculation: \( M = \int_{0}^{1} (12x^2 - 12x^3) \, dx \).

Upon completing the integration process, you'll find the mass \( M = 1 \).

Integration helps us manage calculations that are too complex to be solved directly, especially when dealing with changing densities or irregular shapes.

Calculating mass is a fundamental step in determining the center of mass for non-uniform objects. This step involves integrating the density function over the area bounded by the curves.

First, you need to compute the total mass \( M \) of the plate. Use the integral:\[M = \int_{0}^{1} \delta(x) (x - x^2) \, dx\]This integral considers:

• The density function \( \delta(x) = 12x \).
• The height difference \( x - x^2 \) between the bounding curves \( y = x^2 \) and \( y = x \).

Evaluate this integral by integrating each term separately:

• Integrate \( 12x^2 \) to get \( 4x^3 \).
• Integrate \( -12x^3 \) to get \( -3x^4 \).

Finally, evaluate these from 0 to 1 to find the mass \( M = 1 \). The mass calculation ensures that we're measuring how the total weight of the object is influenced by variable density within the region.