please help me An object occupies the 2D region bounded between the curves yobject at the point (x, y) is given by the function p(x, y)= x* and y = x7. The density of the= x. Find the object's center of mass.

Name: please help me An object occupies the 2D region bounded between the cu
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Description: please help me An object occupies the 2D region bounded between the curves yobject at the point (x, y) is given by the function p(x, y)= x* and y = x7. The density of the= x. Find the object's center of mass.

Given an object bounded by the curves y=x4 and y=x14. The density of the object is given by ρx, y=x.…

Answered: An object occupies the 2D region…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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please help me

An object occupies the 2D region bounded between the curves y
object at the point (x, y) is given by the function p(x, y)
= x* and y = x7. The density of the
= x. Find the object's center of mass.

Expert Solution

Step 1

Given an object bounded by the curves $y = x^{4}$ and $y = x^{\frac{1}{4}}$ .

The density of the object is given by $ρ (x, y) = x$ .

Then the region is plotted as follows,

Here the mass is given by,

$\begin{array}{rcl} M & = & \int_{0}^{1} ρ (x^{\frac{1}{4}} - x^{4}) d x \\ = & \int_{0}^{1} x (x^{\frac{1}{4}} - x^{4}) d x \\ = & \int_{0}^{1} x^{\frac{5}{4}} - x^{5} d x \\ = & {[\frac{x^{\frac{9}{4}}}{\frac{9}{4}} - \frac{x^{6}}{6}]}_{0}^{1} \\ = & \frac{4}{9} - \frac{1}{6} \\ = & \frac{5}{18} \end{array}$

Step 2

Then, moments about x-axis is,

$\begin{array}{rcl} M_{x} & = & \frac{1}{2} \int_{0}^{1} x [{(x^{\frac{1}{4}})}^{2} - {(x^{4})}^{2}] d x \\ = & \frac{1}{2} \int_{0}^{1} x^{\frac{3}{2}} - x^{9} d x \\ = & \frac{1}{2} {[\frac{x^{\frac{5}{2}}}{\frac{5}{2}} - \frac{x^{10}}{10}]}_{0}^{1} \\ = & \frac{1}{2} [\frac{2}{5} - \frac{1}{10}] \\ = & \frac{3}{20} \end{array}$

Also, moments about y-axis is,

$\begin{array}{rcl} M_{y} & = & \int_{0}^{1} x^{2} [x^{\frac{1}{4}} - x^{4}] d x \\ = & \int_{0}^{1} x^{\frac{9}{4}} - x^{6} d x \\ = & {[\frac{x^{\frac{13}{4}}}{\frac{13}{4}} - \frac{x^{11}}{11}]}_{0}^{1} \\ = & [\frac{4}{13} - \frac{1}{11}] \\ = & \frac{31}{143} \end{array}$

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