91Ó°ÊÓ

In mathematics and physics, a density function represents how mass is distributed along an object or region. For a linear wire, the density function \( \delta(x) \) is crucial because it helps us understand how densely packed the mass is in different segments of the wire. In this exercise, the wire's density varies across two intervals.

• From \( x=0 \) to \( x=2 \), the density \( \delta(x) = 4 \), meaning each unit length of this segment is packed with mass equivalent to 4 units.
• From \( x=2 \) to \( x=3 \), the density increases slightly to \( \delta(x) = 5 \).

Using a piecewise function is common in such situations where the density isn't uniform across the entire interval. This allows us to tailor our calculations for the mass and center of mass more accurately over specific regions.

To find the mass of an object with a varying density function, we must integrate the density over the relevant intervals. This process involves dividing the wire into segments where the density is constant, then calculating the mass for each segment. Here's how we do it step-by-step:
- **Calculate Mass for the First Segment:** The density function \( \delta(x) = 4 \) from \( x = 0 \) to \( x = 2 \). The mass \( M_1 \) of this segment is calculated by integrating \( 4 \) over the interval \[ M_1 = \int_{0}^{2} 4 \, dx = 4x \bigg|_0^2 = 8 \].- **Calculate Mass for the Second Segment:** In the next interval, \( x = 2 \) to \( x = 3 \), the density function changes to \( 5 \). The mass \( M_2 \) is \[ M_2 = \int_{2}^{3} 5 \, dx = 5x \bigg|_2^3 = 5 \].- **Total Mass:** Finally, add the masses from both segments to find the total mass \( M \): \[ M = M_1 + M_2 = 8 + 5 = 13 \]. This accumulated mass helps in further calculations, like locating the wire's center of mass.

Integration isn't just a tool for finding areas under curves; it's also powerful for tackling real-world problems like mass distribution and center of mass calculations. For these exercises, integrating the density function over an interval gives us the total mass residing along that interval.
- **Why Use Integration?** When density varies across a length, simple multiplication of length by density isn't enough. Instead, integration sums up infinitely small contributions to mass along the wire, ensuring precision.
- **Integration Steps:** 1. *Choose the Interval:* Define where each density behavior applies, here \( [0, 2] \) and \( (2, 3] \). 2. *Integrate the Density Function:* For each piece, compute the definite integral of the density function over its respective interval: - First segment: \( \int_{0}^{2} 4 \, dx \) - Second segment: \( \int_{2}^{3} 5 \, dx \) 3. *Evaluate the Integrals:* Use basic integration methods to calculate results and add them up for the total mass.
Integrals also help calculate the center of mass by considering distributions with respect to position. This comprehensive calculation is pivotal in designing and analyzing structures depending on balanced weights.