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.
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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.35
35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of
Find the area of the region R.
Verified step by step guidance1
Identify the region R in the first quadrant bounded by the piecewise function \( f(x) \) and the x-axis. The function is given by:
\[
f(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 2 - \sqrt{x} & \text{if } 1 < x \leq 4 \end{cases}
\]
This means the region is bounded above by \( f(x) \) and below by the x-axis from \( x=0 \) to \( x=4 \).
To find the area of region R, set up the integral of \( f(x) \) over the interval \( [0,4] \). Since \( f(x) \) is piecewise, split the integral into two parts:
\[
\text{Area} = \int_0^1 1 \, dx + \int_1^4 (2 - \sqrt{x}) \, dx
\]
Evaluate the first integral:
\[
\int_0^1 1 \, dx = [x]_0^1 = 1 - 0
\]
This represents the area of the rectangle under \( f(x) = 1 \) from 0 to 1.
For the second integral, rewrite the integrand and integrate term-by-term:
\[
\int_1^4 (2 - \sqrt{x}) \, dx = \int_1^4 2 \, dx - \int_1^4 x^{1/2} \, dx
\]
Calculate each integral separately using the power rule for integration.
After finding the antiderivatives, apply the Fundamental Theorem of Calculus by evaluating the definite integrals at the limits \( x=1 \) and \( x=4 \). Then, sum the results of both integrals to get the total area of region R.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Piecewise Functions
A piecewise function is defined by different expressions over distinct intervals of the domain. Understanding how to interpret and work with these functions is essential, especially when calculating areas or integrals, as each piece may require separate treatment.
Recommended video:
05:36
Piecewise Functions
Definite Integrals for Area Calculation
The definite integral of a function over an interval gives the net area between the function's graph and the x-axis. For regions bounded by piecewise functions, the total area is found by summing integrals over each sub-interval where the function is defined differently.
Recommended video:
05:43Definition of the Definite Integral
Square Root Functions and Their Integration
Functions involving square roots, such as √x, require careful integration techniques. Knowing how to rewrite and integrate expressions like 2 - √x is crucial for accurately finding areas under such curves.
Recommended video:
03:29Completing the Square to Rewrite the Integrand Example 7
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).
a. Write a single integral that gives the area of R.
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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.
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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
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 graph of y = 4−x² and the x-axis on the interval [−2,2] is revolved about the line x = −2. What is the volume of the solid that is generated?
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