Question

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The region bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=25\)

Short answer

Answer: The center of mass of the solid is given by \((x_{c}, y_{c}, z_{c}) = (0, 0, z_{c})\), where \(z_{c}\) is calculated using the integral: \[\frac{1}{V} \int_{0}^{2\pi}\int_{0}^{5}\int_{\rho^{2}}^{25} z\rho dz \, d\rho d\theta\] and \(V\) is the volume, calculated by: \[\int_{0}^{2\pi}\int_{0}^{5}\int_{\rho^{2}}^{25} \rho dz d\rho d\theta.\]

Step-by-step solution

Find the Limits of Integration

We know that the solid is bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=25\). We need to find the limits of integration in cylindrical coordinates (\(\rho, \theta, z\)). The equation of the paraboloid can be rewritten in cylindrical coordinates as \(z=\rho^{2}\). The limits of integration will be: - \(\rho\): from 0 to 5 (as \(\rho=\sqrt{25}=5\)) - \(\theta\): from 0 to \(2\pi\) (since it is rotationally symmetric) - \(z\): from \(\rho^{2}\) to 25

Calculate the Volume

We will now find the volume, \(V\), of the solid by integrating the region using cylindrical coordinates. To do this, we will integrate the function \(1\) multiplied by the jacobian, \(\rho\), with the found limits of integration. $$V=\int_{0}^{2\pi}\int_{0}^{5}\int_{\rho^{2}}^{25} \rho dz d\rho d\theta$$ Now, we will find the z-coordinate of the centroid, \(z_{c}\). |Integration Setup for z-coordinate of Centroid|

Find the Limits of Integration

The limits of integration for the z-coordinate of the centroid, \(z_{c}\), are the same as for the volume: - \(\rho\): from 0 to 5 - \(\theta\): from 0 to \(2\pi\) - \(z\): from \(\rho^{2}\) to 25

Calculate the z-coordinate of Centroid

Now, we will find the z-coordinate of the centroid, \(z_{c}\), by integrating the function \(z\) multiplied by the jacobian, \(\rho\), with the limits of integration we found above, and dividing by the volume, \(V\): $$z_{c} = \frac{1}{V} \int_{0}^{2\pi}\int_{0}^{5}\int_{\rho^{2}}^{25} z\rho dz \, d\rho d\theta$$

Finally, Find the Center of Mass

Using the symmetry of the solid and the calculations for the z-coordinate of the centroid, we find the center of mass: $$\text{Center of Mass} = (x_{c}, y_{c}, z_{c}) = (0,0, z_{c})$$

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that is particularly useful for dealing with problems involving symmetry around a central axis. Unlike Cartesian coordinates, which use \(x\), \(y\), and \(z\) to specify a point in space, cylindrical coordinates use \(\rho\), \(\theta\), and \(z\).

• \(\rho\) (rho) represents the radial distance from the z-axis outwards to the point.
• \(\theta\) (theta) is the angle around the z-axis, often measured in radians.
• \(z\) remains the same as in Cartesian coordinates, indicating height or depth.

This coordinate system is especially helpful for problems involving rotational symmetry. In this case, because our region is bounded by a paraboloid and a horizontal plane, using cylindrical coordinates simplifies the process of integration.

Paraboloid

A paraboloid is a three-dimensional surface that can be visualized as a 3D parabola extended in all directions. In mathematical terms, paraboloids are defined using quadratic equations. For example, the formula \(z = x^2 + y^2\) describes a paraboloid that opens upwards. This kind of surface is called an elliptic paraboloid.
A key property of a paraboloid is its curvature, which makes it suitable for applications involving focusing and collecting, such as satellite dishes or reflectors. In this problem, the solid bounded by the paraboloid and a plane creates a region that, when sliced horizontally, forms circular cross-sections. These sections imply rotational symmetry, making cylindrical coordinates an ideal choice for calculating the mass distribution.

Integration Limits

When calculating volumes or areas, setting the correct integration limits is crucial. These limits define the extent of the region over which the integration is performed.
For our problem involving the paraboloid \(z = x^2 + y^2\) and the plane \(z = 25\), it's helpful to reformulate these equations into cylindrical coordinates:

• \(z = \rho^2\) is the equation of the paraboloid in cylindrical form.
• The limits for \(\rho\) are from 0 to 5, determined by setting \(\rho^2 = 25\).
• \(\theta\) extends from 0 to \(2\pi\), due to the full circular symmetry.
• The \(z\)-value ranges between \(\rho^2\) and 25, acknowledging the paraboloid’s lower boundary and the plane’s upper limitation.

Setting these precise limits allows for accurate integration across the defined solid region.

Symmetry in Calculus

Symmetry in calculus, particularly in problems involving integration, allows for significant simplification in mathematical calculations. When a shape is symmetric, certain characteristics repeat, meaning that we can predict or reduce calculations based on known patterns.
For our solid—bounded by a rotationally symmetric paraboloid and a horizontal plane—exploiting symmetry is a practical choice. The rotational symmetry implies that the x and y components of the center of mass are zero, reducing the need for complex calculations for \(x_c\) and \(y_c\).

• Symmetry ensures that the center of mass lies along the axis of rotation.
• This property simplifies the problem to calculating only the z-coordinate of the centroid \(z_c\).
• Using symmetry allows for efficient determination of the center of mass with minimal computation.

Understanding and leveraging symmetry is an essential skill in calculus, making complex systems more manageable.