Book cover for Calculus Early Transcendental Functions

Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

ISBN #9781285774770

6th Edition

8,973 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers applications of integration in computing areas between curves and volumes of solids of revolution. Key points include setting up proper integrals by determining the upper and lower boundaries or outer and inner radii, using integration as a process of accumulation, and carefully dividing regions when curves intersect. The methods not only apply to academic exercises but also to real-world engineering, economics, and design problems.

Learning Objectives

1

Explain how to set up and evaluate definite integrals to find the area of a region between two curves.

2

Describe integration as an accumulation process and apply it to geometric problems.

3

Determine the points of intersection between curves to correctly set up integrals for regions with overlapping boundaries.

4

Apply the disk (and washer) method to calculate the volume of solids of revolution.

5

Distinguish between vertical and horizontal representative elements when formulating integration problems.

Key Concepts

CONCEPT

DEFINITION

Area Between Two Curves

The area of a region bounded by two curves f(x) and g(x) on an interval [a, b] is given by the integral ∫[a,b] (f(x) - g(x)) dx, where f(x) ≥ g(x) along the interval.

Representative Rectangle

A small element (rectangle or disk) used in the Riemann sum approximation of an area or volume which, when integrated, accumulates to the total area or volume.

Integration as an Accumulation Process

The concept of summing up infinitely many infinitesimal quantities (such as area or volume) to obtain a total value via limits; the definite integral is the limit of a Riemann sum.

Disk Method

A technique for finding the volume of a solid of revolution by slicing the solid perpendicular to the axis of revolution, treating each slice as a disk with volume π(R(x))² dx.

Washer Method

A modification of the disk method used when the solid of revolution has a hole; the volume is calculated by subtracting the volume of the inner disk from the outer disk using ∫ π[(R_outer(x))² - (R_inner(x))²] dx.

Solid of Revolution

A three-dimensional object obtained by revolving a plane region about a line (axis) in the same plane.

Example Problems

Example 1

Set up the definite integral that gives the area of the region. (GRAPH CAN'T COPY) $$\begin{aligned} &y_{1}=x^{2}-6 x\\ &y_{2}=0 \end{aligned}$$

Example 2

Set up the definite integral that gives the area of the region. (GRAPH CAN'T COPY) $$\begin{aligned} &y_{1}=x^{2}+2 x+1\\ &y_{2}=2 x+5 \end{aligned}$$

Example 3

Set up the definite integral that gives the area of the region. (GRAPH CAN'T COPY) $$\begin{aligned} &y_{1}=x^{2}-4 x+3\\ &y_{2}=-x^{2}+2 x+3 \end{aligned}$$

Example 4

Set up the definite integral that gives the area of the region. (GRAPH CAN'T COPY) $$\begin{aligned} &y_{1}=x^{2}\\ &y_{2}=x^{3} \end{aligned}$$

Example 5

Set up the definite integral that gives the area of the region. (GRAPH CAN'T COPY) $$\begin{aligned} &y_{1}=3\left(x^{3}-x\right)\\ &y_{2}=0 \end{aligned}$$
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Step-by-Step Explanations

QUESTION

Given two continuous functions f(x) and g(x) on [a, b] with f(x) ≥ g(x), how do you find the area of the region between them?

STEP-BY-STEP ANSWER:

Step 1: Identify the curves f(x) and g(x) and the interval [a, b] over which they are defined or intersect.
Step 2: Verify which function is the upper curve (f(x)) and which is the lower curve (g(x)) on [a, b].
Step 3: Write the integrand as the difference f(x) - g(x).
Step 4: Set up the definite integral A = ∫ from a to b (f(x) - g(x)) dx.
Step 5: Evaluate the integral to obtain the area between the curves.
Final Answer: The area is A = ∫[a,b] (f(x) - g(x)) dx.

Area Between Two Curves

QUESTION

How do you compute the volume of a solid of revolution generated by revolving a region about an axis using the disk method?

STEP-BY-STEP ANSWER:

Step 1: Identify the function or boundary curve that defines the region to be revolved and determine the axis of revolution.
Step 2: Determine the radius R(x) of the representative disk formed by a slice perpendicular to the axis.
Step 3: Express the volume of an infinitesimally thin disk as dV = π (R(x))² dx.
Step 4: Set up the integral for the volume as V = ∫ from a to b π (R(x))² dx over the interval of interest.
Step 5: Evaluate the integral to find the total volume of the solid.
Final Answer: The volume is V = π ∫[a,b] (R(x))² dx.

Volume Using the Disk Method

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Common Mistakes

  • Failing to correctly identify which function is on top, resulting in a negative area or incorrect integral set-up.
  • Neglecting to find all points of intersection when the curves cross each other multiple times, leading to an incomplete or erroneous area calculation.
  • Incorrectly setting the limits of integration especially when using horizontal representative rectangles instead of vertical slices (or vice versa).
  • Mixing up the formulas for the disk method and the washer method, particularly when an inner radius is involved.
  • Overlooking the need to split the integral into separate parts when the relationship between the curves changes within the domain.