Explanation Given: Density ρ = 1 Parabola y = 2 − 3 x 2 and below by the line 3 x + 2 y = 1 . Calculation: Consider a lamina with ρ = 1 over the region bounded above by parabola y = 2 − 3 x 2 and below the line 3 x + 2 y = 1 Find the centroid for this region Centroid is provided by x ¯ = M y m And, y ¯ = M x m So first calculate the mass m and moment of mass M x , M y Put 2 − 3 x 2 = 1 − 3 x 2 to know where line 3 x + 2 y = 1 and parabola y = 2 − 3 x 2 intersect 2 − 3 x 2 = 1 − 3 x 2 4 − 6 x 2 = 1 − 3 x 6 x 2 − 3 x − 3 = 0 6 x 2 − 6 x + 3 x − 3 = 0 Further solving 6 x 2 − 6 x + 3 x − 3 = 0 6 x ( x − 1 ) + 3 ( x − 1 ) = 0 ( x − 1 ) ( 6 x + 3 ) = 0 x = 1 , − 1 2 In the given region when x varies from − 1 2 to 1, y varies from 2 − 3 x 2 to 1 − 3 x 2 So to find m evaluate the integral m = ∫ − 1 2 1 ∫ 2 − 3 x 2 1 − 3 x 2 ρ d y d x m = ∫ − 1 2 1 ∫ 2 − 3 x 2 1 − 3 x 2 ρ d y d x = ∫ − 1 2 1 ∫ 2 − 3 x 2 1 − 3 x 2 1 d y d x = ∫ − 1 2 1 ( ∫ 2 − 3 x 2 1 − 3 x 2 d y ) d x = ∫ − 1 2 1 ( y | 2 − 3 x 2 1 − 3 x 2 ) d x Putting the limits m = ∫ − 1 2 1 ( y | 2 − 3 x 2 1 − 3 x 2 ) d x = ∫ − 1 2 1 ( 1 2 − 3 x 2 − 2 + 3 x 2 ) d x = 3 ∫ − 1 2 1 x 2 d x − 3 2 ∫ − 1 2 1 x d x − 3 2 ∫ − 1 2 1 d x = ( x 3 | − 1 2 1 ) − 3 4 ( x 2 | − 1 2 1 ) − 3 2 ( x | − 1 2 1 ) Putting the limits m = ( x 3 | − 1 2 1 ) − 3 4 ( x 2 | − 1 2 1 ) − 3 2 ( x | − 1 2 1 ) = 1 + 1 8 − 3 4 ( 1 − 1 4 ) − 3 2 ( 1 + 1 2 ) = 9 8 − 9 16 − 9 4 = − 27 16 Now find M x M x = ∫ − 1 2 1 ∫ 2 − 3 x 2 1 − 3 x 2 y d y d x = ∫ − 1 2 1 ( ∫ 2 − 3 x 2 1 − 3 x 2 y d y ) d x = 1 2 ∫ − 1 2 1 ( y 2 | 2 − 3 x 2 1 − 3 x 2 ) d x = 1 2 ∫ − 1 2 1 ( 1 4 ( 1 + 9 x 2 − 6 x ) − ( 4 + 9 x 4 − 12 x 2 ) ) d x Further solving M x = 1 2

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

See similar textbooks

Videos

Question

Chapter 12.6, Problem 9PS

To determine

To find: the centroid for the regions of the solid

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.6, Problem 8PS

Chapter 12.6, Problem 10PS

Students have asked these similar questions

3 If the pollution index of a point (x, y) is given by p(x, y) = x². Calculate the total pollution of the region bounded by y = x² and y = 2x. Find the total pollution and the centre of pollution in this region.

Problems: 1. Find the centroid for the area surrounded by the curve y=x', y-axis and the line y=8 ? 2. Find the centroid of the area enclosed between the line y=10-2x and the two Cartesian axis? 3. The area surrounded by the curve y=2x²-x+6, y-axis and the line y=9 represents a part of a solid area .Find its centroid? 4. Find the centroid of the shape shown in figure? 7 4 4 4

4. Perform the following (a) A gold-plated plate D is defined by the rectangle enclosed by the lines x = 0, x = 2π, y = 0, and y = π (in centimeters) and has a mass density of 8(x, y) = y² sin²(4x) + 2 grams per square centimeter. If the gold costs 7 USD per gram, how much is the gold present on the plate worth? (b) Find the average value of the function f(x, y, z) = e¯² over the sphere x² + y² + z² ≤ 1.

Chapter 12 Solutions

Calculus

Ch. 12.1 - Prob. 1PS Ch. 12.1 - Prob. 2PS Ch. 12.1 - Prob. 3PS Ch. 12.1 - Prob. 4PS Ch. 12.1 - Prob. 5PS Ch. 12.1 - Prob. 6PS Ch. 12.1 - Prob. 7PS Ch. 12.1 - Prob. 8PS Ch. 12.1 - Prob. 9PS Ch. 12.1 - Prob. 10PS

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

the centroid for the regions of the solid

Solution Summary: The author explains how to find the centroid for the regions of the solid.

Videos

To find: the centroid for the regions of the solid

Want to see the full answer?

Chapter 12 Solutions