91Ó°ÊÓ

Triple integration is a powerful mathematical tool used to calculate volume, mass, and other quantities in three-dimensional spaces. Imagine having a solid object, and we wish to find its mass or another property that extends through its entire volume.

In spherical coordinates, which are especially useful for objects with symmetry around a point, we define our position using three parameters: \(\rho\) (the distance from the origin), \(\phi\) (the angle down from the positive z-axis), and \(\theta\) (the angle in the xy-plane from the positive x-axis). When performing triple integration in spherical coordinates, each differential element of volume \(dV\) is expressed as \(\rho^2 \sin\phi \, d\rho \, d\phi \, d\theta\). This factor accounts for the volume of a tiny wedge-shaped piece of the solid.

When setting up the integration for the solid described in the exercise, limits for \(\rho\) are determined by the surfaces, \(x^2 + y^2 + z^2 = z\) and \(x^2 + y^2 + z^2 = R^2\), where it ranges from \(\cos\phi\) to \(R\). \(\phi\) operates from 0 to \(\pi/2\) due to the upper hemisphere, and \(\theta\) naturally spans 0 to \(2\pi\) to cover the full rotation.

• Triple integration allows us to sum up influences over the full volume of the object.
• In this example, we first integrate with respect to \(\rho\), then \(\phi\), and finally \(\theta\).

A density function is a crucial concept when calculating the mass of a distributed object. It tells us how much mass exists per unit volume throughout the solid. For example, in this exercise, the density function is \(\rho(x, y, z)=\frac{1}{\sqrt{x^2 + y^2 + z^2}}\). This function shows that density decreases as you move away from the origin, reflecting a realistic physical scenario where parts of a material further out are less dense.

To find the mass of a solid with a variable density, you multiply the density function by differential volume elements and integrate over the region of interest. In our case, the density function multiplies with the volume element \(\rho \sin\phi\), derived from spherical coordinates, adjusting to \(\frac{1}{\rho}\) since \(\rho(x, y, z)\) simplifies in spherical terms.

• The density function gives important insights into how mass is spread in space.
• By integrating the product of the density and the volume element, we calculate the total mass.

The mass of a solid is obtained by integrating its density function over the entire volume of the solid. Here, the goal is to find \(R\) such that the mass is \(\frac{7 \pi}{2}\).

Through the integration process, we first tackled \(\int_{\cos \phi}^{R} \rho \, d\rho\), giving us the expression \(\frac{1}{2}(R^2 - \cos^2 \phi)\). Continuing this method, integrating over \(\phi\) yields another expression after substituting and simplifying, resulting in \(\frac{1}{2} R^2 - \frac{1}{8}\).

Finally, integrating over \(\theta\) ensures we account for the volume through the full rotation around the z-axis. The last step requires setting up the equation: \( (\frac{1}{2} R^2 - \frac{1}{8})2\pi = \frac{7\pi}{2}\), which simplifies to solve for \(R\). The found \(R\) gives the exact radius needed for the solid to have the specified mass of \(\frac{7 \pi}{2}\).

• To find the mass, integrate the density function over the given volume limits.
• The solution involves solving the mass equation for the parameter \(R\).