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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 6.6.27a
Capitolo 6, Problema 6.6.27a

Consider the following curves on the given intervals.  


a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis. 


y=tan x , for 0≤x≤π/4; about the x-axis 

Guida verificata passo dopo passo
1
Identify the formula for the surface area generated by revolving a curve about the x-axis. The surface area \( S \) is given by the integral \( S = \int_a^b 2\pi y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \), where \( y = f(x) \).
Given the curve \( y = \tan x \) on the interval \( 0 \leq x \leq \frac{\pi}{4} \), substitute \( y = \tan x \) into the formula.
Compute the derivative \( \frac{dy}{dx} = \sec^2 x \).
Substitute \( y = \tan x \) and \( \frac{dy}{dx} = \sec^2 x \) into the surface area integral to get \( S = \int_0^{\frac{\pi}{4}} 2\pi \tan x \sqrt{1 + (\sec^2 x)^2} \, dx \).
Simplify the expression inside the square root as much as possible before setting up the integral for evaluation.

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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 that accounts for the curve's length and distance from the axis. For a function y = f(x) revolved about the x-axis, the surface area is given by the integral of 2π times the radius (y) times the arc length element.
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Example 1: Minimizing Surface Area

Arc Length Element (ds)

The differential arc length element ds represents a small segment of the curve and is calculated as ds = √(1 + (dy/dx)^2) dx. This accounts for both horizontal and vertical changes along the curve, ensuring the surface area integral accurately measures the curved surface.
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Percorso guidato
06:29
Arc Length of Parametric Curves

Trigonometric Function Differentiation

To set up the integral, you need the derivative of y = tan x, which is sec^2 x. Understanding how to differentiate trigonometric functions is essential to find dy/dx, which is used in the arc length formula for the surface area calculation.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
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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.

a. On what intervals is the object moving in the positive direction?

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Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


a. Graph the velocity function for both riders. 

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Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.

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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 = cos 2x, for 0 ≤ x ≤ π

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Region R is revolved about the line x=4 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point y in [1, 3]?

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A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.


a. Use the shell method to write an integral for the volume of the torus.

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