Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

Calculus

9. Graphical Applications of Integrals

Introduction to Volume & Disk Method

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1:18 minutes

Problem 6.3.1

Textbook Question

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).

Verified step by step guidance

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 solid has a circular base; cross sections perpendicular to the base are squares. What method should be used to find the volume of the solid?

Textbook Question

Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.

a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].

Textbook Question

Why is the disk method a special case of the general slicing method?

Textbook Question

Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures).

c. Write an integral for the volume of the solid.

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