Textbook Question
Determine the area of the shaded region in the following figures.
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 6.7.13
13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.
ρ(x)=1+sin x, for 0≤x≤π
Verified step by step guidance1
Identify the given density function \( \rho(x) = 1 + \sin x \) and the interval over which the bar extends, which is \( 0 \leq x \leq \pi \).
Recall that the mass \( M \) of a one-dimensional object with density function \( \rho(x) \) over the interval \( [a, b] \) is given by the integral \( M = \int_a^b \rho(x) \, dx \).
Set up the integral for the mass using the given density and interval: \( M = \int_0^{\pi} (1 + \sin x) \, dx \).
Break the integral into two simpler integrals: \( M = \int_0^{\pi} 1 \, dx + \int_0^{\pi} \sin x \, dx \).
Evaluate each integral separately: the integral of 1 over \( [0, \pi] \) and the integral of \( \sin x \) over \( [0, \pi] \), then sum the results to find the total mass.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Density Function
The density function ρ(x) describes how mass is distributed along the length of the bar. It gives the mass per unit length at each point x, allowing us to calculate total mass by integrating over the interval.
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Properties of Functions
Definite Integral for Mass
The total mass of a one-dimensional object with variable density is found by integrating the density function over the given interval. Specifically, mass = ∫ from a to b of ρ(x) dx, summing all infinitesimal mass elements.
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Integration of Trigonometric Functions
Since the density function includes sin x, understanding how to integrate trigonometric functions is essential. The integral of sin x over an interval can be computed using standard antiderivatives, facilitating the calculation of total mass.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Set up a sum of two integrals that equals the area of the shaded region bounded by the graphs of the functions f and g on [a, c] (see figure). Assume the curves intersect at x=b.
Textbook Question
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = (x²+2)^3/2 / 3 on [0, 1]
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Textbook Question
13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.
ρ(x) = {x² if 0≤x≤1 {x(2-x) if 1<x≤2
Textbook Question
Work from force How much work is required to move an object from x=0 to x=3 (measured in meters) in the presence of a force (in N) given by F(x)=2x acting along the x-axis?
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Textbook Question
Drinking juice A glass has circular cross sections that taper (linearly) from a radius of 5 cm at the top of the glass to a radius of 4 cm at the bottom. The glass is 15 cm high and full of orange juice. How much work is required to drink all the juice through a straw if your mouth is 5 cm above the top of the glass? Assume the density of orange juice equals the density of water.