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
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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 6.5.19
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)
Verified step by step guidance1
Rewrite the given curve equation in terms of \( y \): \( x = 2y - 4 \). Since \( x \) is expressed as a function of \( y \), we will use the arc length formula with respect to \( y \).
Recall the arc length formula for a curve \( x = f(y) \) from \( y = a \) to \( y = b \):
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy \]
Calculate the derivative \( \frac{dx}{dy} \) from the given function:
\[ \frac{dx}{dy} = \frac{d}{dy}(2y - 4) = 2 \]
Substitute \( \frac{dx}{dy} = 2 \) into the arc length integral and set the limits from \( y = -3 \) to \( y = 4 \):
\[ L = \int_{-3}^{4} \sqrt{1 + (2)^2} \, dy = \int_{-3}^{4} \sqrt{1 + 4} \, dy = \int_{-3}^{4} \sqrt{5} \, dy \]
Evaluate the integral by factoring out the constant \( \sqrt{5} \):
\[ L = \sqrt{5} \int_{-3}^{4} dy = \sqrt{5} [y]_{-3}^{4} = \sqrt{5} (4 - (-3)) = \sqrt{5} \times 7 \]
This gives the arc length in terms of \( \sqrt{5} \) and the interval length.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula
The arc length of a curve defined by a function can be found using the integral formula: L = ∫√(1 + (dy/dx)²) dx or L = ∫√(1 + (dx/dy)²) dy, depending on the variable of integration. This formula sums infinitesimal line segments along the curve to find its total length.
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Arc Length of Parametric Curves
Parametric and Implicit Differentiation
When the curve is given in terms of y or implicitly, it is important to correctly find the derivative dx/dy or dy/dx. This may involve rearranging the equation or using implicit differentiation to express the derivative needed for the arc length formula.
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06:49Differentiation of Parametric Curves
Geometric Interpretation and Verification
After calculating the arc length using calculus, it is useful to verify the result geometrically if possible. For linear or simple curves, the arc length corresponds to the distance between endpoints, providing a way to check the correctness of the integral calculation.
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04:18Geometric Sequences - Recursive Formula
Related Practice
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.
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Textbook Question
3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.
y = 2 cos 3x on [−π,π]
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Textbook Question
46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.
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Textbook Question
The region R is bounded by the graph of f(x)=2x(2−x) and the x-axis. Which is greater, the volume of the solid generated when R is revolved about the line y=2 or the volume of the solid generated when R is revolved about the line y=0? Use integration to justify your answer.
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