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Ch. 6 - Applications of Definite Integrals
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 6 - Applications of Definite IntegralsProblem 6.PE.19
Chapter 6, Problem 6.PE.19

Find the lengths of the curves in Exercises 19–22.
y = x¹/² ― (1/3) x³/² , 1 ≤ x ≤ 4

Verified step by step guidance
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Identify the function given: \(y = x^{\frac{1}{2}} - \frac{1}{3} x^{\frac{3}{2}}\) and the interval for \(x\) is \(1 \leq x \leq 4\).
Recall the formula for the length of a curve \(y = f(x)\) from \(x = a\) to \(x = b\): \[L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
Find the derivative \(\frac{dy}{dx}\) of the function: \[\frac{dy}{dx} = \frac{d}{dx} \left(x^{\frac{1}{2}} - \frac{1}{3} x^{\frac{3}{2}}\right)\] Use the power rule to differentiate each term.
Square the derivative to get \(\left(\frac{dy}{dx}\right)^2\) and then add 1 inside the square root: \[1 + \left(\frac{dy}{dx}\right)^2\]
Set up the integral for the arc length: \[L = \int_{1}^{4} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\] This integral can then be evaluated (analytically or numerically) to find the length of the curve.

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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 found using the integral L = ∫_a^b √(1 + (dy/dx)²) dx. This formula calculates the distance along the curve by summing infinitesimal line segments, accounting for both horizontal and vertical changes.
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Arc Length of Parametric Curves

Derivative of the Function

To apply the arc length formula, you must first find the derivative dy/dx of the given function y = x^(1/2) - (1/3)x^(3/2). This involves using power rule differentiation to handle fractional exponents accurately.
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Derivatives of Other Trig Functions

Evaluating Definite Integrals

After substituting dy/dx into the arc length integral, you evaluate the definite integral from x = 1 to x = 4. This may require algebraic simplification or numerical methods if the integral is complex or does not have a simple antiderivative.
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Definition of the Definite Integral
Related Practice
Textbook Question

Areas of Surfaces of Revolution

In Exercises 23–26, find the areas of the surfaces generated by revolving the curves about the given axes.

_____

y = √2x + 1 , 0 ≤ x ≤ 3 ; x-axis"

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Textbook Question

Work

Pumping a conical tank A right-circular conical tank, point down, with top radius 5 ft and height 10 ft, is filled with a liquid whose weight-density is 60lb/ft³. How much work does it take to pump the liquid to a point 2 ft above the tank? If the pump is driven by a motor rated at 275ft-lb/sec (1/2 hp), how long will it take to empty the tank? 

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Textbook Question

Centers of Mass and Centroids

Find the center of mass of a thin, flat plate covering the region enclosed by the parabola 𝔂² = 𝓍 and the line 𝓍 = 2𝔂 if the density function is δ(𝔂) = 1 + 𝔂. (Use horizontal strips.)

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Textbook Question

Volumes

Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x⁴ , and the lines x = 1 and x = ―1 about

a. the x-axis

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Textbook Question

Work

Earth’s attraction The force of attraction on an object below Earth’s surface is directly proportional to its distance from Earth’s center. Find the work done in moving a weight of w lb located α mi below Earth’s surface up to the surface itself. Assume Earth’s radius is a constant r mi. 

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Textbook Question

Work


Lifting equipment A rock climber is about to haul up 100 N (about 22.5 lb) of equipment that has been hanging beneath her on 40 m of rope that weighs 0.8 N/m. How much work will it take? (Hint: Solve for the rope and equipment separately, then add.)

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