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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 6.5.24a
Capitolo 6, Problema 6.5.24a

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 = x³/3, for −1≤x≤1

Guida verificata passo dopo passo
1
Recall the formula for the arc length of a curve defined by a function \( y = f(x) \) on the interval \( [a, b] \): \[ L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
Identify the function and the interval: here, \( y = \frac{x^3}{3} \) and \( x \) ranges from \( -1 \) to \( 1 \).
Compute the derivative \( \frac{dy}{dx} \) of the function: \[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{x^3}{3} \right) = x^2 \]
Substitute \( \frac{dy}{dx} = x^2 \) into the arc length formula to get the integral: \[ L = \int_{-1}^1 \sqrt{1 + (x^2)^2} \, dx = \int_{-1}^1 \sqrt{1 + x^4} \, dx \]
This integral expression represents the arc length of the curve on the given interval. It can be evaluated using a calculator or numerical methods since it does not have a simple antiderivative.

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Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral ∫ from a to b of √(1 + (dy/dx)²) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.
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Percorso guidato
06:29
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 = x³/3. Differentiating gives dy/dx = x², which is then squared inside the integral to find the integrand √(1 + (dy/dx)²).
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06:30
Derivatives of Other Trig Functions

Setting up and Simplifying the Integral

After finding dy/dx, substitute it into the arc length integral and simplify the expression under the square root. For y = x³/3, the integral becomes ∫ from -1 to 1 of √(1 + x⁴) dx, which may require numerical methods or a calculator for evaluation.
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Percorso guidato
08:12
Integration by Parts for Definite Integrals Example 7
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Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days). 

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Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.


a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

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13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


a. Determine when the motion is in the positive direction and when it is in the negative direction. 


v(t) = 50e^−2t on [0, 4]

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Region R is revolved about the line y=1 to form a solid of revolution.


a. What is the radius of a cross section of the solid at a point x in [0, 4]?

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Consider the region R in the first quadrant bounded by y=x^1/n and y=x^n, where n>1 is a positive number.


a. Find the volume V(n) of the solid generated when R is revolved about the x-axis. Express your answer in terms of n.

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A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.

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