Textbook Question
Emptying a cylindrical tank A cylindrical water tank has height 8 m and radius 2m (see figure).
b. Is it true that it takes half as much work to pump the water out of the tank when it is half full as when it is full? Explain.
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 6.5.35b
Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.
b. ∫a^b √1+36 cos² 2xdx
Verified step by step guidance1
Recall the formula for the arc length of a differentiable function \( y = f(x) \) on the interval \([a,b]\):
\[
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]
Compare the given integral for arc length:
\[
\int_a^b \sqrt{1 + 36 \cos^2(2x)} \, dx
\]
with the general formula. This means:
\[
1 + \left(\frac{dy}{dx}\right)^2 = 1 + 36 \cos^2(2x)
\]
From the equality above, isolate \( \frac{dy}{dx} \):
\[
\left(\frac{dy}{dx}\right)^2 = 36 \cos^2(2x)
\]
which implies
\[
\frac{dy}{dx} = \pm 6 \cos(2x)
\]
Integrate \( \frac{dy}{dx} = \pm 6 \cos(2x) \) with respect to \( x \) to find the family of functions:
\[
y = \pm 6 \int \cos(2x) \, dx + C
\]
Recall that \( \int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C \), so the family of functions is:
\[
y = \pm 3 \sin(2x) + C
\]
where \( C \) is an arbitrary constant.
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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 differentiable function y = f(x) on [a, b] is given by the integral ∫_a^b √(1 + (f'(x))²) dx. This formula measures the length of the curve by summing infinitesimal line segments, where the integrand involves the derivative of the function.
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Arc Length of Parametric Curves
Relating the Integrand to the Derivative
To find functions with a given arc length integral, identify the expression inside the square root as 1 + (f'(x))². Equate this to the given integrand to solve for f'(x), which helps determine the family of functions whose derivatives satisfy the condition.
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04:16Intro To Related Rates
Solving Differential Equations for Families of Functions
Once f'(x) is found, integrate it to obtain the general form of f(x). Since integration introduces an arbitrary constant, the solution represents a family of functions, not a unique one, matching the problem's requirement for multiple solutions.
Recommended video:
06:06Solving Separable Differential Equations
Related Practice
Textbook Question
Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.
b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position
Textbook Question
Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).
b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.
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Textbook Question
Two runners At noon (t=0), Alicia starts running along a long straight road at 4 mi/hr. Her velocity decreases according to the function v(t) = 4 / t + 1 for t≥0. At noon, Boris also starts running along the same road with a 2-mi head start on Alicia; his velocity is given by u(t) = 2 / t + 1, for t≥0. Assume t is measured in hours.
b. When, if ever, does Alicia overtake Boris?
Textbook Question
A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.
b. Use the washer method to write an integral for the volume of the torus.
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.