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 area of each of the regions R₁,R₂, and R₃.
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.R.23
14–25. {Use of Tech} Areas of regions Determine the area of the given region.
The region in the first quadrant bounded by y = x/6 and y = 1−|x/2−1|
Verified step by step guidance1
First, identify the curves that bound the region in the first quadrant: the line \(y = \frac{x}{6}\) and the piecewise function \(y = 1 - \left| \frac{x}{2} - 1 \right|\).
Next, analyze the function \(y = 1 - \left| \frac{x}{2} - 1 \right|\) by considering the expression inside the absolute value. Set \(\frac{x}{2} - 1 = 0\) to find the critical point \(x = 2\) where the function changes its form.
Rewrite \(y = 1 - \left| \frac{x}{2} - 1 \right|\) as a piecewise function:
For \(x \leq 2\), \(y = 1 - (1 - \frac{x}{2}) = \frac{x}{2}\),
For \(x > 2\), \(y = 1 - (\frac{x}{2} - 1) = 2 - \frac{x}{2}\).
Determine the points of intersection between \(y = \frac{x}{6}\) and each piece of the piecewise function:
- For \(x \leq 2\), solve \(\frac{x}{6} = \frac{x}{2}\),
- For \(x > 2\), solve \(\frac{x}{6} = 2 - \frac{x}{2}\).
These intersection points will serve as limits of integration.
Set up the integral(s) for the area of the region in the first quadrant bounded by the curves. The area can be found by integrating the difference between the upper and lower functions over the appropriate intervals determined by the intersection points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Understanding Piecewise and Absolute Value Functions
The function y = 1 − |x/2 − 1| involves an absolute value, which creates a piecewise definition. Understanding how to rewrite and interpret absolute value expressions as piecewise linear functions is essential to identify the shape and boundaries of the region.
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Piecewise Functions
Finding Points of Intersection
Determining the area bounded by two curves requires finding their points of intersection. Solving y = x/6 and y = 1 − |x/2 − 1| simultaneously helps establish the limits of integration for calculating the area.
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04:50Critical Points
Definite Integration to Find Area Between Curves
The area between two curves is found by integrating the difference of the upper and lower functions over the interval defined by their intersection points. Setting up and evaluating the definite integral accurately yields the desired area.
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05:23Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question
58–61. Arc length Find the length of the following curves.
y = 2x+4 on [−2,2] (Use calculus.)
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Textbook Question
Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).
b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.
Textbook Question
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?
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Textbook Question
Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:
a. Apply the disk method and integrate with respect to y.
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.
Use the shell method to find an integral, or sum of integrals, that equals the volume of the solid obtained by revolving region R₃ about the line x=3. Do not evaluate the integral.
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