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Chapter 6: Problem 30

True-False Determine whether the statement is true or false. Explain your answer. [In these exercises, assume that a solid \(S\) of volume \(V\) is bounded by two parallel planes perpendicular to the \(x\) -axis at \(x=a\) and \(x=b\) and that for each \(x\) in \([a, b], A(x)\) denotes the cross-sectional area of \(S\) perpendicular to the \(x\) -axis.] $$ \begin{aligned} &\text { The average value of } A(x) \text { on the interval }[a, b] \text { is given by }\\\ &V /(b-a) . \end{aligned} $$

Short Answer

Expert verified
The statement is true.

Step by step solution

01

Understand the Problem Statement

The problem asks us to determine if the statement "The average value of \( A(x) \) on the interval \([a, b]\) is given by \( V/(b-a) \)" is true or false, where \( S \) is a solid bounded by planes, with volume \( V \), and \( A(x) \) is the cross-sectional area perpendicular to the \( x \)-axis.
02

Recall the Formula for Average Value of a Function

The average value of a function \( A(x) \) over an interval \([a,b]\) is computed as \( \frac{1}{b-a} \int_{a}^{b} A(x) \, dx \). In this context, \( A(x) \) represents the cross-sectional area at each point \( x \) within the interval \([a,b]\).
03

Relate the Average Value Formula to the Volume

The volume \( V \) of the solid \( S \) is given by \( \int_{a}^{b} A(x) \, dx \). Hence, the average value of \( A(x) \) can be expressed as \( \frac{1}{b-a} \times \int_{a}^{b} A(x) \, dx = \frac{V}{b-a} \).
04

Compare the Derived Expression with the Problem Statement

We have shown that the expression for the average value of \( A(x) \) is \( \frac{V}{b-a} \), which matches exactly with the statement provided in the problem. Therefore, the statement is logically consistent with the mathematical formula for average value of a function.
05

Conclusion

Since our calculation matches the given statement, the statement is true. The average value of the cross-sectional area \( A(x) \) over the specified interval is indeed \( \frac{V}{b-a} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume of Solid
When we talk about the volume of a solid, we often refer to the space enclosed within its boundaries. For a solid bounded by two parallel planes perpendicular to the x-axis at points \( x=a \) and \( x=b \), we can determine its volume using calculus. The volume \( V \) of such a solid \( S \) is calculated by integrating its cross-sectional area function \( A(x) \) along the interval from \( a \) to \( b \). This gives us the formula: \[ V = \int_{a}^{b} A(x) \, dx \]. This integration process essentially "sums up" infinitely small slices of the solid from \( x=a \) to \( x=b \).
Key points to remember about volume of a solid bounded by planes:
  • It's found by integrating the cross-sectional area over the given interval.
  • The planes at \( x=a \) and \( x=b \) serve as boundaries.
  • Knowing \( A(x) \) at different \( x \)-values helps compute the overall volume.
Cross-Sectional Area
Understanding the cross-sectional area is crucial to finding the volume of complex solids. Imagine slicing a loaf of bread; each slice is a cross-section of the loaf. Similarly, for a solid \( S \) bounded by planes perpendicular to the x-axis, \( A(x) \) represents the area of these "slices" at different points along the x-axis.
Each value of \( A(x) \) gives us the area of a slice of the solid at a particular \( x \)-coordinate. By knowing \( A(x) \) for every \( x \) within the interval \([a, b]\), we get a complete picture of the changing shape of the solid's cross sections along its length.
A few important aspects of cross-sectional area include:
  • \( A(x) \) varies with \( x \), reflecting the solid's shape.
  • The integral of \( A(x) \) over \( [a, b] \) gives the solid's volume.
  • Finding \( A(x) \) helps visualize and calculate the solid's properties.
Integration in Calculus
Integration is a fundamental concept in calculus used to compute areas under curves, which directly helps in finding volumes of solids. In the context of volumes, integrating the cross-sectional area \( A(x) \) across an interval helps you "add up" all the tiny slices of the solid to find the total volume.
The integration process for finding volume can be broken down into:
  • Identifying the function \( A(x) \) that represents the cross-sectional area at any x-value.
  • Setting up the definite integral \( \int_{a}^{b} A(x) \, dx \) to compute the total volume from x = a to x = b.
  • Calculating the integral to find the total volume \( V \) enclosed by the solid.
Integration allows us to efficiently compute volumes even when exact geometric formulas do not exist for complex shapes.
Remember these essential points about integration in calculating volumes:
  • Integration aggregates all cross-sections over the interval.
  • It's a powerful tool for finding areas and volumes.
  • It requires knowing the function defining the area or cross-section.

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