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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 6.5.14
Chapter 6, Problem 6.5.14

9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = x^3/2 / 3 − x^1/2 on [4, 16]

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1
Identify the function given: \(y = \frac{x^{3/2}}{3} - x^{1/2}\), and the interval \([4, 16]\) over which we want to find the arc length.
Recall the formula for the arc 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: First, rewrite the function as \(y = \frac{1}{3} x^{3/2} - x^{1/2}\). Then differentiate term-by-term using the power rule.
Square the derivative \(\left(\frac{dy}{dx}\right)^2\) and add 1 inside the square root to form the integrand: \(\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\)
Set up the definite integral for the arc length over the interval \([4, 16]\): \(L = \int_4^{16} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\). Evaluate this integral to find the arc length.

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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 sums the lengths of infinitesimal line segments along the curve, providing the total distance traveled along it.
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Arc Length of Parametric Curves

Derivative of the Function

To apply the arc length formula, you must compute the derivative dy/dx of the given function y = (x^(3/2))/3 − x^(1/2). This involves using power rule differentiation for fractional exponents to find the slope of the curve at any point.
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Derivatives of Other Trig Functions

Definite Integration

After finding the integrand √(1 + (dy/dx)^2), evaluate the definite integral from x = 4 to x = 16. This requires integrating the expression over the interval to obtain the exact arc length, often involving substitution or numerical methods if the integral is complex.
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Definition of the Definite Integral
Related Practice
Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.


y=4−x^2,x=2, and y=4; about the y-axis

Textbook Question

35–38. Shell and washer methods Let R be the region bounded by the following curves. Use both the shell method and the washer method to find the volume of the solid generated when R is revolved about the indicated axis.


y = (x−2)³ −2,x=0, and y=25; about the y-axis

Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 


x = x³ ,y = 1, and x = 0; about the x-axis

Textbook Question

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).


a(t) = −32; v(0)=50; s(0)=0

Textbook Question

Find the area of the region described in the following exercises.


The region bounded by y=√x, y=2x−15, and y=0

Textbook Question

46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.