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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.37
Chapter 6, Problem 6.PE.37

Centers of Mass and Centroids

Find the centroid of a thin, flat plate covering the region enclosed by the parabolas 𝔂 = 2𝓍² and 𝔂 = 3 ― 𝓍² .

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Identify the region bounded by the curves \( y = 2x^2 \) and \( y = 3 - x^2 \). The centroid \( (\bar{x}, \bar{y}) \) of the region can be found by first determining the points of intersection of these two curves.
Set the two functions equal to find the intersection points: \( 2x^2 = 3 - x^2 \). Solve for \( x \) to find the limits of integration for the region.
Calculate the area \( A \) of the region by integrating the difference between the upper curve and the lower curve over the interval found: \[ A = \int_{a}^{b} \left( (3 - x^2) - (2x^2) \right) \, dx = \int_{a}^{b} (3 - 3x^2) \, dx \].
Find the coordinates of the centroid using the formulas: \[ \bar{x} = \frac{1}{A} \int_{a}^{b} x \left( (3 - x^2) - (2x^2) \right) \, dx = \frac{1}{A} \int_{a}^{b} x (3 - 3x^2) \, dx \] and \[ \bar{y} = \frac{1}{2A} \int_{a}^{b} \left( (3 - x^2)^2 - (2x^2)^2 \right) \, dx \].
Evaluate the integrals and simplify the expressions to find \( \bar{x} \) and \( \bar{y} \), which give the coordinates of the centroid of the plate.

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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 average position of all points in a plane figure. For a thin, flat plate with uniform density, it corresponds to the point where the plate would balance perfectly. It is found by calculating the average x and y coordinates weighted by the area.
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Area Between Curves

To find the centroid, you first need the area of the region bounded by the given curves. This involves integrating the difference between the upper and lower functions over the interval where they intersect, which gives the total area of the enclosed region.
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Moments About the Axes

The coordinates of the centroid are found using moments: the moment about the y-axis gives the x-coordinate, and the moment about the x-axis gives the y-coordinate. These moments are integrals of x or y multiplied by the area element, divided by the total area.
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Related Practice
Textbook Question

Work


Assume that a spring does not follow Hooke’s Law. Instead, the force required to stretch the spring x ft from its natural length is Ζ’(𝓍) = 10𝓍³/Β² lb . How much work does it take to

a. stretch the spring 4 ft from its natural length?

Textbook Question

Volumes

Volume of a solid sphere hole A round hole of radius √3 ft is bored through the center of a solid sphere of radius 2 ft. Find the volume of material removed from the sphere.

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Textbook Question

Volumes

Find the volumes of the solids in Exercises 1–18.

The solid lies between planes perpendicular to the x-axis at x = 0 and x = 4. The cross-sections of the solid perpendicular to the x-axis between these planes are circular disks whose diameters run from the curve xΒ² = 4y to the curve yΒ² = 4x.

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Textbook Question

Volumes

Find the volumes of the solids in Exercises 1–18.

The solid lies between planes perpendicular to the x-axis at x = 0 and x = 1. The cross-sections perpendicular to the x-axis between these planes are circular disks whose diameters run from the parabola y = x² to the parabola y = √x.

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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/ftΒ³. 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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