Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.3.1

Chapter 6, Problem 6.3.1

Suppose a cut is made through a solid object perpendicular to the x-axis at a particular point x. Explain the meaning of A(x).

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Understand that when we talk about a cut made through a solid object perpendicular to the x-axis at a point \( x \), we are referring to a cross-sectional slice of the object at that position.

The notation \( A(x) \) represents the area of this cross-sectional slice at the specific point \( x \) along the x-axis.

In other words, \( A(x) \) is a function that assigns to each value of \( x \) the area of the shape you get when you slice the solid perpendicular to the x-axis at that point.

This concept is fundamental in calculus when calculating volumes of solids using methods such as the disk, washer, or cross-sectional area methods, where the volume is found by integrating \( A(x) \) over an interval.

So, \( A(x) \) essentially captures how the size of the cross-section changes as you move along the x-axis through the solid.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cross-Sectional Area

A(x) represents the area of the cross-section of the solid object at the position x. It is the two-dimensional slice obtained by cutting the object perpendicular to the x-axis, showing the shape and size of the object at that point.

Function of a Variable

A(x) is a function that assigns a numerical value (the area) to each position x along the x-axis. Understanding that A depends on x helps analyze how the cross-sectional area changes as you move along the object.

Application in Volume Calculation

Knowing A(x) is essential for finding the volume of the solid using integration. By integrating A(x) with respect to x over an interval, you sum up all cross-sectional areas to determine the total volume.

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Textbook Question

A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where 0≤h≤8 0 ≤ ℎ ≤ 8 . Find the volume of water in the bowl as a function of h. (Check the special cases h=0 and h=8.)

Textbook Question

Use the general slicing method to find the volume of the following solids.

The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares

Textbook Question

Surface area of a cone Find the surface area of a cone (excluding the base) with radius 4 and height 8 using integration and a surface area integral.

Textbook Question

Find the area of the following regions, expressing your results in terms of the positive integer n≥2.

Let Aₙ be the area of the region bounded by f(x)=x^1/n and g(x)=x^n on the interval [0,1], where n is a positive integer. Evaluate lim n→∞ Aₙ and interpret the result. br

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Textbook Question

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.

ρ(x) = 5e^-2x,for 0≤x≤4

Textbook Question

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.

Find the volume of the solid obtained by revolving region R₂ about the y-axis.

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