Textbook Question
Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y=f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.
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 6.5.9
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = −8x−3 on [−2, 6] (Use calculus.)
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Identify the function given: \(y = -8x - 3\) and the interval \([-2, 6]\) over which we want to find the arc length.
Recall the formula for the arc length \(L\) of a curve \(y = f(x)\) from \(x = a\) to \(x = b\):
\(L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\)
Compute the derivative \(\frac{dy}{dx}\) of the function \(y = -8x - 3\). Since this is a linear function, find \(\frac{dy}{dx}\) explicitly.
Substitute \(\frac{dy}{dx}\) into the arc length formula to get the integrand \(\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\), which will be a constant expression because the derivative is constant.
Evaluate the definite integral \(\int_{-2}^{6} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\) by integrating the constant integrand over the interval length \((6 - (-2))\).
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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 y = f(x) from x = a to x = b is found using the integral formula L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula sums the lengths of infinitesimal line segments along the curve, providing the total distance traveled along it.
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Arc Length of Parametric Curves
Derivative of the Function
To apply the arc length formula, you need the derivative dy/dx of the function y = f(x). The derivative represents the slope of the curve at any point, and its square is used inside the square root to account for the curve's steepness in the length calculation.
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06:30Derivatives of Other Trig Functions
Definite Integration
Calculating arc length requires evaluating a definite integral over the given interval [a, b]. This process involves integrating the expression √(1 + (dy/dx)^2) with respect to x, which may require algebraic simplification or substitution to solve.
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
Assume f and g are continuous, with f(x) ≥ g(x) ≥ 0 on [a, b]. The region bounded by the graphs of f and g and the lines x=a and x=b is revolved about the y-axis. Write the integral given by the shell method that equals the volume of the resulting solid.
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Textbook Question
9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)
The work required to empty the top half of the tank
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Textbook Question
Leaky Bucket A 1-kg bucket resting on the ground contains 3 kg of water. How much work is required to raise the bucket vertically a distance of 10 m if water leaks out of the bucket at a constant rate of 1/5 kg/m? Assume the weight of the rope used to raise the bucket is negligible. (Hint: Use the definition of work, W = ∫a^bF(y) dy, where F is the variable force required to lift an object along a vertical line from y=a to y=b.)
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Textbook Question
For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.
R is bounded by y=1−x^3, the x-axis, and the y-axis.
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Textbook Question
Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given line.
x=2−secy,x=2,y=π/3, and y=0; about x=2
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