Textbook Question
Finding arc length
Find the length of the curve
y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.AAE.22
Centroid of a region
Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.
Verified step by step guidance1
Identify the region bounded by the curves and lines. The curves are given by \(y = \pm (1 - x^2)^{-1/2}\), and the vertical boundaries are \(x = 0\) and \(x = 1\). This region is symmetric about the x-axis because of the \(\pm\) sign.
Set up the expressions for the area \(A\) of the region. Since the region is symmetric about the x-axis, the total area can be found by integrating the upper curve and doubling it: \(A = 2 \int_0^1 (1 - x^2)^{-1/2} \, dx\).
Write the formulas for the coordinates of the centroid \((\bar{x}, \bar{y})\). Because of symmetry about the x-axis, \(\bar{y} = 0\). The \(x\)-coordinate of the centroid is given by \(\bar{x} = \frac{1}{A} \int_0^1 x \cdot 2(1 - x^2)^{-1/2} \, dx\).
Evaluate the integrals for the area \(A\) and the moment integral for \(\bar{x}\). Use an appropriate substitution such as \(x = \sin \theta\) to simplify the integrals involving \((1 - x^2)^{-1/2}\).
After computing the integrals symbolically, substitute the results back into the formula for \(\bar{x}\) to express the centroid coordinates as \((\bar{x}, 0)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Centroid of a Region
The centroid is the geometric center or 'balance point' of a planar region. It is found by calculating the average position of all points in the region, typically using integrals to find the coordinates (x̄, ȳ). For regions bounded by curves, the centroid coordinates are given by the moments of area divided by the total area.
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Area of Polar Regions
Definite Integrals for Area and Moments
Definite integrals are used to compute the area under curves and the moments needed to find the centroid. The area is found by integrating the difference between the bounding functions, while moments involve integrating the product of the coordinate and the function. These integrals are evaluated over the given interval, here from x = 0 to x = 1.
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05:43Definition of the Definite Integral
Handling Curves Defined by y = ±(1 − x²)^(-1/2)
The curves y = ±(1 − x²)^(-1/2) represent the upper and lower branches of a function with vertical asymptotes at x = ±1. Understanding their behavior and domain is crucial for setting up correct integrals. Since the region is bounded between x = 0 and x = 1, the integrals must consider the infinite behavior near x = 1 carefully.
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06:29Arc Length of Parametric Curves
Related Practice
Textbook Question
18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:
a. the y-axis.
Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ cot⁶(2x) dx
Textbook Question
Finding volume
The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.
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Textbook Question
Evaluate the limits in Exercise 7 and 8.
lim (x → ∞) ∫₋ˣ^ˣ sin t dt
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Textbook Question
Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫ cos t dt / (1 - cos t)