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.
(b) Find the mass of the right half of the rod (5 ≤ x ≤ 10) .

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Step 1: Understand the problem. The mass of the rod is the area under the density curve. We are tasked with finding the mass of the right half of the rod, which corresponds to the interval 5 ≤ x ≤ 10.

Step 2: Analyze the graph. The density function is piecewise linear. From x = 5 to x = 6, the density increases linearly from 5 g/cm to 6 g/cm. From x = 6 to x = 10, the density remains constant at 6 g/cm.

Step 3: Break the interval into two subintervals: (a) x = 5 to x = 6, where the density is a linear function, and (b) x = 6 to x = 10, where the density is constant. Calculate the area under the curve for each subinterval separately.

Step 4: For the interval x = 5 to x = 6, the area is the area of a trapezoid. Use the formula for the area of a trapezoid: A = (1/2) * (base1 + base2) * height. Here, base1 = 5, base2 = 6, and height = 1 (difference in x-values).

Step 5: For the interval x = 6 to x = 10, the area is the area of a rectangle. Use the formula for the area of a rectangle: A = width * height. Here, width = 4 (difference in x-values) and height = 6 (density value). Add the areas from both subintervals to find the total mass of the right half of the rod.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Density Function

A density function describes how mass is distributed along a given length. In this case, the density of the rod varies with its length, 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 (5 ≤ x ≤ 10). 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 over a specific interval. For this problem, it involves evaluating the integral of the density function from x = 5 to x = 10. The result gives the total mass of the right half of the rod, reflecting the varying density across that segment.