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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.6.23d
Capitolo 6, Problema 6.6.23d

Determine whether the following statements are true and give an explanation or counterexample. 


d. Let f(x)=12x^2.. The area of the surface generated when the graph of f on [−4, 4] is revolved about the y-axis is twice the area of the surface generated when the graph of f on [0, 4] is revolved about the y-axis.

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Identify the function and the interval: We have \( f(x) = 12x^2 \) and two intervals: \([-4,4]\) and \([0,4]\). We are revolving the graph about the y-axis to find the surface area generated.
Recall the formula for the surface area of a curve revolved about the y-axis: \[ S = \int_a^b 2\pi x \sqrt{1 + \left(\frac{df}{dx}\right)^2} \, dx \]. Here, \(x\) is the radius from the y-axis, and \(f(x)\) is the height.
Compute the derivative \( \frac{df}{dx} \) for \( f(x) = 12x^2 \): \[ \frac{df}{dx} = 24x \]. Substitute this into the surface area integral.
Set up the two surface area integrals: - For \([-4,4]\): \[ S_1 = \int_{-4}^4 2\pi x \sqrt{1 + (24x)^2} \, dx \] - For \([0,4]\): \[ S_2 = \int_0^4 2\pi x \sqrt{1 + (24x)^2} \, dx \].
Analyze the integrand's behavior over \([-4,4]\): Since \(x\) appears as a factor and \(x\) is negative on \([-4,0)\), the integrand is negative there, so the integral over \([-4,4]\) is not simply twice the integral over \([0,4]\). This suggests the statement is false because the surface area integral is not symmetric about zero due to the \(x\) term.

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Surface Area of Revolution

The surface area generated by revolving a curve around an axis is found using an integral formula involving the function, its derivative, and the radius from the axis of revolution. For revolution about the y-axis, the radius is the x-value, and the formula accounts for the curve's length and distance from the axis.
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09:07
Example 1: Minimizing Surface Area

Symmetry of Functions and Intervals

When a function is even (symmetric about the y-axis), the graph on [−a, a] is symmetric. This symmetry affects integrals over symmetric intervals, often allowing simplification by doubling the integral from [0, a]. However, for surface areas revolving around an axis, the radius term may break this symmetry.
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Percorso guidato
06:21
Properties of Functions

Effect of Radius in Surface Area Integrals

In surface area calculations revolving around the y-axis, the radius is the x-coordinate, which is negative on [−a, 0] and positive on [0, a]. Since radius appears as an absolute value or squared term, the contribution from negative x-values may differ, impacting whether the surface area over [−a, a] is exactly twice that over [0, a].
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09:07
Example 1: Minimizing Surface Area
Pratica correlata
Domanda del libro di testo

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

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Domanda del libro di testo

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the x-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

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Domanda del libro di testo

Displacement and distance from velocity Consider the velocity function shown below of an object moving along a line. Assume time is measured in seconds and distance is measured in meters. The areas of four regions bounded by the velocity curve and the t-axis are also given.

d. What is the displacement of the object over the interval [0, 8]? 

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Domanda del libro di testo

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.

d. A piecewise function for s(t)

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Domanda del libro di testo

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.


d. How long does it take the racer to travel 300 ft?

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Displacement and distance from velocity Consider the graph shown in the figure, which gives the velocity of an object moving along a line. Assume time is measured in hours and distance is measured in miles. The areas of three regions bounded by the velocity curve and the t-axis are also given.

d. What is the displacement of the object over the interval [0,5]?

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