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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.5.24a
Chapter 6, Problem 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

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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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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 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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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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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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Integration by Parts for Definite Integrals Example 7
Related Practice
Textbook Question

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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Textbook Question

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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Textbook Question

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.

a. ∫a^b √1+16x⁴ dx

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Textbook Question

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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Textbook Question

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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Textbook Question

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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