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Ch. 6 - Applications of Definite Integrals
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 6 - Applications of Definite IntegralsProblema 6.PE.19
Capitolo 6, Problema 6.PE.19

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

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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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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
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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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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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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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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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Work

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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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