Textbook Question
Mass of two bars Two bars of length L have densities ρ₁(x) = 4e^−x and ρ₂(x) = 6e^−2x, for 0≤x≤L.
a. For what values of L is bar 1 heavier than bar 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 6.6.23a
Determine whether the following statements are true and give an explanation or counterexample.
a. If the curve y=f(x) on the interval [a, b] is revolved about the y-axis, the area of the surface generated is ∫f(b)f(a) 2πf(y)√1+f′(y)^2 dy.
Verified step by step guidance1
Step 1: Understand the problem context. The problem involves finding the surface area generated by revolving a curve about the y-axis. The curve is given by y = f(x) on the interval [a, b]. We need to verify if the given integral expression for the surface area is correct.
Step 2: Recall the formula for the surface area when revolving a curve about the y-axis. If the curve is expressed as x = g(y) (i.e., x as a function of y), then the surface area S is given by:
\(S = \int_{c}^{d} 2\pi x \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy\)
where [c, d] is the interval for y.
Step 3: Note that the problem gives the curve as y = f(x), so to use the formula revolving around the y-axis in terms of y, we need to express x as a function of y, i.e., find the inverse function x = f^{-1}(y). Then, the derivative \(\frac{dx}{dy}\) is the derivative of the inverse function.
Step 4: Compare the given integral \(\int_{f(b)}^{f(a)} 2\pi f(y) \sqrt{1 + (f'(y))^2} \, dy\) with the correct formula. Notice that the integrand uses \(f(y)\) and \(f'(y)\), but since f is originally a function of x, \(f(y)\) and \(f'(y)\) do not make sense unless f is inverted. This suggests the given integral is not correctly formulated.
Step 5: Conclusion: The given integral is not the correct formula for the surface area generated by revolving y = f(x) about the y-axis. The correct approach requires expressing x as a function of y and using the formula involving \(x\) and \(\frac{dx}{dy}\). Therefore, the statement is false.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Surface Area of Revolution
The surface area generated by revolving a curve around an axis is found by integrating the circumference of circular slices times the arc length element. For revolution about the y-axis, the formula involves integrating 2π times the radius (x or function of y) multiplied by the differential arc length along y.
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Example 1: Minimizing Surface Area
Parametrization and Variable of Integration
When revolving a curve about the y-axis, the integral is typically expressed in terms of y, requiring the function to be rewritten as x = g(y). The limits of integration correspond to y-values, and the derivative inside the integral must be with respect to y, not x.
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Guided course
08:02Parameterizing Equations
Arc Length Element in Terms of y
The differential arc length element ds is given by √(1 + (dx/dy)^2) dy when integrating with respect to y. Using f'(y) instead of dx/dy or mixing variables leads to incorrect formulas, so careful differentiation and substitution are essential.
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06:29Arc Length of Parametric Curves
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution.
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Textbook Question
Calculating work for different springs Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke’s law is obeyed.
a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position
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Textbook Question
21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = ln x, for 1≤x≤4
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Textbook Question
For the given regions R₁ and R₂, complete the following steps.
a. Find the area of region R₁.
R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.
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Textbook Question
9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
a. The displacement between t=0 and t=5
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