Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.
(a) Find the mass of the left half of the rod (0 ≤ x ≤ 5) .

Verified step by step guidance

1. Step 1: Understand the problem. The mass of the rod is the area under the density curve. For the left half of the rod (0 ≤ x ≤ 5), we need to calculate the area under the curve from x = 0 to x = 5.
2. Step 2: Analyze the graph. The density function is piecewise linear. From x = 0 to x = 2, the density is constant at 2 g/cm. From x = 2 to x = 5, the density increases linearly from 2 g/cm to 5 g/cm.
3. Step 3: Break the area into two regions. Region 1 is a rectangle from x = 0 to x = 2 with height 2 g/cm. Region 2 is a trapezoid from x = 2 to x = 5 with bases 2 g/cm and 5 g/cm and height 3 cm.
4. Step 4: Calculate the area of Region 1. The area of a rectangle is given by \( \text{Area} = \text{length} \times \text{height} \). Here, \( \text{length} = 2 \) cm and \( \text{height} = 2 \) g/cm.
5. Step 5: Calculate the area of Region 2. The area of a trapezoid is given by \( \text{Area} = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height} \). Here, \( \text{base}_1 = 2 \) g/cm, \( \text{base}_2 = 5 \) g/cm, and \( \text{height} = 3 \) cm.

Key Concepts

Density Function

A density function describes how mass is distributed over a given length. In this case, the density of the rod varies along its length, which is represented graphically. Understanding the density function is crucial for calculating the mass, as it provides the necessary values to integrate over the specified interval.

Integration

Integration is a fundamental concept in calculus used to find the area under a curve. In this context, the mass of the rod can be determined by integrating the density function over the specified interval (0 to 5 cm for the left half). This process allows us to sum up the infinitesimal contributions of mass along the length of the rod.

Definite Integral

A definite integral calculates the accumulation of quantities, such as area or mass, over a specific interval. For this problem, the definite integral of the density function from 0 to 5 cm will yield the total mass of the left half of the rod. It is essential to set up the integral correctly based on the piecewise nature of the density function shown in the graph.