Textbook Question
{Use of Tech} Decreasing velocity A projectile is fired upward, and its velocity in m/s is given by v(t) = 200e^−t/10, for t≥0.
a. Graph the velocity function, for t≥0.
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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.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
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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Textbook Question
Force on a dam Find the total force on the face of a semicircular dam with a radius of 20 m when its reservoir is full of water. The diameter of the semicircle is the top of the dam.
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Textbook Question
Comparing volumes Let R be the region bounded by y=1/x^p and the x-axis on the interval [1, a], where p>0 and a>1 (see figure). Let Vₓ and Vᵧ be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively.
c. Find a general expression for Vₓ in terms of a and p. Note that p=1/2 is a special case. What is Vₓ when p=1/2?
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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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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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