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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.6.59a
Chapter 8, Problem 8.6.59a

Centroid:
Find the centroid of the region cut from the first quadrant by the curve
y = 1/√(x + 1) and the line x = 3.

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Identify the region bounded by the curve \(y = \frac{1}{\sqrt{x + 1}}\), the vertical line \(x = 3\), and the coordinate axes in the first quadrant. The region lies between \(x = 0\) and \(x = 3\), and above the \(x\)-axis.
Set up the integral for the area \(A\) of the region using the formula \(A = \int_0^3 y \, dx = \int_0^3 \frac{1}{\sqrt{x + 1}} \, dx\). This will give the total area under the curve from \(x=0\) to \(x=3\).
Find the coordinates of the centroid \((\bar{x}, \bar{y})\) using the formulas: \(\bar{x} = \frac{1}{A} \int_0^3 x y \, dx = \frac{1}{A} \int_0^3 x \frac{1}{\sqrt{x + 1}} \, dx\) and \(\bar{y} = \frac{1}{2A} \int_0^3 y^2 \, dx = \frac{1}{2A} \int_0^3 \left(\frac{1}{\sqrt{x + 1}}\right)^2 \, dx\).
Evaluate each integral separately: the area integral, the \(x\)-moment integral \(\int_0^3 x y \, dx\), and the \(y\)-moment integral \(\int_0^3 y^2 \, dx\). Use substitution if necessary, for example, let \(u = x + 1\) to simplify the integrals.
After computing the integrals, substitute the results back into the centroid formulas to find \(\bar{x}\) and \(\bar{y}\). These values give the coordinates of the centroid of the region.

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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 plane region. It is found by calculating the average position of all points in the area, typically using integrals to find the coordinates (x̄, ȳ). For regions bounded by curves, the centroid coordinates are given by the moments divided by the area.
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Area of Polar Regions

Definite Integrals for Area and Moments

Definite integrals are used to compute the area under a curve and the moments about the axes. The area is found by integrating the function over the given interval, while moments involve integrating the product of the function and x or y coordinates. These integrals are essential to determine the centroid coordinates.
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Definition of the Definite Integral

Region Bounded by a Curve and a Vertical Line

The region is defined by the curve y = 1/√(x + 1), the vertical line x = 3, and the coordinate axes in the first quadrant. Understanding the limits of integration and the shape of the region is crucial for setting up the correct integrals to find area and moments.
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Finding Area Between Curves on a Given Interval
Related Practice
Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 2 to 4 of 1/(s - 1)² ds

Textbook Question

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.

a. About the y-axis.

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Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.

a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?

1
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Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 1 to 3 of (2x - 1) dx

Textbook Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 0 to 3 of 1/√(x + 1) dx

Textbook Question

Finding volume: Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, about

a. The y-axis.

(See Exercise 57 for a graph.)