Question

Find the center of mass of the following plane region with variable density ellipse...

A) Integral calculus
B) Vector calculus
C) Analytic geometry
D) Multivariable calculus

Answer 1

To find the center of mass for various shapes with either uniform or variable density, multivariable calculus is the primary mathematical discipline used, often involving techniques from vector calculus, analytic geometry, and integral calculus.

Step-by-step explanation:

To find the center of mass of a plane region with variable density, such as an ellipse, we would primarily use multivariable calculus. In a scenario where we have a rectangular material with nonuniform density given by p(x, y) = poxy, and we are asked to find the center of mass, multivariable calculus is used to set up and evaluate the required integrals. The process typically involves calculating the moments about the x and y axes and then dividing by the total mass of the object.

To find the center of mass of more complex shapes such as a cube with a part removed, a cone of uniform density, or a semicircular plate, the same principles apply but with adjustments to account for the shapes and densities involved. For systems of particles, techniques from vector calculus and analytic geometry are used to determine the weighted average position of all particles. In practice, the subject matter can touch upon concepts from integral calculus as well, particularly when dealing with continuous mass distributions.

Answer 2

The center of mass of the given plane region with variable density ellipse can be determined using multivariable calculus. Thus the D option is right answer.

Step-by-step explanation:

To find the center of mass of a plane region with variable density, we use the concept of double integration in multivariable calculus. Consider an ellipse in the xy-plane described by the equation
`\((x^2)/(a^2) + (y^2)/(b^2) = 1\)`, where
`\(a\)`and
`\(b\)` represent the semi-major and semi-minor axes, respectively. The region's density is given by a function
`\(\delta(x, y)\)`.

The formula to compute the coordinates of the center of mass
`(\( \bar{x}, \bar{y}\))` for a region
`\(D\)` with variable density is given by:

`\[ \bar{x}`=
`(\iint_D x \delta(x, y) \, dA)/(\iint_D \delta(x, y) \, dA) \]`

`\[ \bar{y} = (\iint_D y \delta(x, y) \, dA)/(\iint_D \delta(x, y) \, dA) \]`

Here,
`\(dA\)` represents an infinitesimal area element in the region
`\(D\)`. To evaluate these integrals, we set up the integral bounds corresponding to the ellipse's area in the xy-plane and compute the double integrals for both numerator and denominator.

We integrate over the ellipse region by using appropriate bounds for
`\(x\)`and
`\(y\) (typically \(x = -a\) to \(x = a\)` and
`\(y = -b \sqrt{1 - (x^2)/(a^2)}\)`to
`\(y = b \sqrt{1 - (x^2)/(a^2)}\))`and integrate the density function
`\(\delta(x, y)\)` times
`\(x\)` and
`\(y\)`separately. Finally, dividing these integrals by the total area integral
`\(\iint_D \delta(x, y) \, dA\)` gives the coordinates of the center of mass
`(\( \bar{x}, \bar{y}\))` for the ellipse region with variable density.