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Chapter 6: Problem 1

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

Short Answer

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Question: Explain the meaning of A(x) when a cut is made through a solid object perpendicular to the x-axis at a particular point x. Answer: A(x) represents the cross-sectional area of a solid object at a particular point x along the x-axis when a cut is made perpendicular to the x-axis. It helps in understanding the shape and size of the object at that specific x location. This concept is significant in calculus for calculating properties such as volume and mass of 3-dimensional solid objects.

Step by step solution

01

Definition of A(x)

A(x) represents the cross-sectional area of a solid object at a particular point x along the x-axis. When a cut is made perpendicular to the x-axis, A(x) describes the area of the resulting section.
02

Identifying the cut and plane

Begin by recognizing that a cut through the solid object is made perpendicular to the x-axis, at a specific point x. This implies that the cut creates a plane at that point, and this plane is parallel to the yz-plane.
03

Cross-sectional area

The cross-sectional area is the area of the section created by the cut or plane. In this case, A(x) is the cross-sectional area of the solid object at the point x. It can be thought of as a "slice" of the object at that specific location.
04

Visualizing A(x)

It might be helpful to visualize A(x) as the shadow of the object cast by a light source located at an infinite distance along the x-axis. As the x-value varies, the cross-sectional area will change according to the shape and size of the object at that particular x location.
05

Importance of A(x) in calculus

A(x) is commonly used in problems involving calculus, particularly when dealing with volume or mass of 3-dimensional solid objects. By finding an equation or function for A(x), it enables us to calculate the volume of the object by integrating A(x) with respect to x over the range of interest. In conclusion, A(x) represents the cross-sectional area of a solid object at a particular point x along the x-axis when a cut is made perpendicular to it. It is a significant concept in calculus and is useful for calculating properties such as volume and mass when dealing with 3-dimensional objects.

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