The boundary of a lamina consists of the semicircles yV1- x2 and y- V36 - x2 together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is proportional to its distance from the origin. (x, y) = 0,
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![**Center of Mass Calculation of a Variously Shaped Lamina**
The problem explores the calculation of the center of mass of a lamina, whose boundary consists of two semicircles and the portions of the x-axis that join them. The semi-circles are defined by the equations:
\[ y = \sqrt{1 - x^2} \]
\[ y = \sqrt{36 - x^2} \]
These semicircles are combined with sections of the x-axis to form the composite shape.
Additionally, the density at any point in the lamina is proportional to its distance from the origin. Given this setup, the center of mass, expressed as \((\bar{x}, \bar{y})\), needs to be determined.
From the calculation provided, the center of mass of the lamina is given by:
\[ (\bar{x}, \bar{y}) = \left( 0, \frac{7}{\pi} \right) \]
This solution implies that the center of mass lies on the y-axis. The y-coordinate is highlighted by a red error indicator, suggesting careful verification is needed. However, the provided coordinate accurately represents the center of mass based on the parameters and given calculations.
For further educational insight, it is recommended to explore the methods of integrating to find centers of mass under variable density conditions and the steps involved in solving similar problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ba0aec5-65e2-43b8-bb6a-70de8eae6b73%2F2d0da6ac-15b2-4eb9-a8b2-468a5b706858%2Fwfsipvm_processed.jpeg&w=3840&q=75)
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