Problem 30 - Advanced Engineering Mathematics

Question

Set up, but do not evaluate, the iterated integrals giving the mass of the solid that has the given shape and density. $$ x^{2}+y^{2}-z^{2}=1, z=-1, z=2 ; p(x, y, z)=z^{2} $$

Step-by-Step Solution

Answer

The iterated integral is \( \int_{-1}^2 \int_0^{2\pi} \int_0^{\sqrt{z^2+1}} z^2 r \, dr \, d\theta \, dz \).

Step 1: Understand the Region of Integration

The solid is bounded by the hyperboloid given by \( x^2 + y^2 - z^2 = 1 \) and planes \( z = -1 \) and \( z = 2 \). This implies the region of integration is a truncated solid within these surfaces.

Step 2: Identify the Density Function

The density function given in the problem is \( \rho(x, y, z) = z^2 \). This density function will be integrated to find the mass of the solid.

Step 3: Set Up the Boundaries for Integration

Since the equation \( x^2 + y^2 - z^2 = 1 \) is equivalent to \( x^2 + y^2 = z^2 + 1 \) for a fixed \( z \), the solid's boundary can be rewritten. The new limits for \( x \) and \( y \) being a circle with a radius \( \sqrt{z^2+1} \).

Step 4: Choose a Suitable Order of Integration

To ease computations, use cylindrical coordinates: \( x = r\cos(\theta) \), \( y = r\sin(\theta) \), \( z = z \). This yields \( x^2 + y^2 = r^2 \), thus \( r^2 = z^2 + 1 \) leads to \( r = \sqrt{z^2 + 1} \).

Step 5: Set Up the Iterated Integral

Write the mass integral as: \[ \int_{-1}^2 \left(\int_0^{2\pi} \left(\int_0^{\sqrt{z^2+1}} (z^2) r \, dr\right) d\theta\right) dz \]This represents integrating over the volume, across \( r \), \( \theta \), and \( z \), with the function \( z^2 \cdot r \) (taking into account the Jacobian \( r \) in cylindrical coordinates).

Key Concepts

Density Function

In physics and mathematics, a density function describes how mass is distributed within an object or region. In simplest terms, it tells us how much material is packed in a specific area. For the given problem, our density function is represented as \( \rho(x, y, z) = z^2 \). This means the density varies with the square of the \( z \)-coordinate.

To compute the mass, one must integrate the density over the volume of the object. This gives the total mass as it accounts for variations in density throughout the solid. In this problem, the density is higher where \( z \) is larger due to the \( z^2 \) term, implying more material concentration at higher \( z \) values. Integrating this density function within the given boundaries ultimately provides the mass of the solid.

Cylindrical Coordinates

Cylindrical coordinates offer a useful way to express locations in three-dimensional space using three parameters: \( r \), \( \theta \), and \( z \).

- **\( r \)**: The radial distance from the z-axis, analogous to the radius in polar coordinates.- **\( \theta \)**: The angle measured from the positive x-axis in the xy-plane.- **\( z \)**: The height above or below the xy-plane, straightforward as in Cartesian coordinates.

Cylindrical coordinates simplify problems involving symmetrically shaped objects like cylinders and circles because they match the natural symmetry of these forms. In the exercise, the transformation to cylindrical coordinates, \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \), results in \[ x^2 + y^2 = r^2 \], making calculations more manageable. The bounds for \( r \) are determined by the original equation \( x^2 + y^2 = z^2 + 1 \), thus setting \( r = \sqrt{z^2 + 1} \) ensures the integration occurs within the region.

Volume of a Solid

The volume of a solid is essentially the amount of three-dimensional space it occupies. For complicated shapes, calculus, especially iterated integrals, enables us to find this volume precisely.

To find the volume using integration, we first express our region of interest in terms of parameters suitable for integration. With the use of cylindrical coordinates in the setup problem, the region's boundaries are clear: - For \( z \), between -1 and 2 due to the planes \( z = -1 \) and \( z = 2 \).- For \( \theta \), spanning a full circle from 0 to \( 2\pi \).- For \( r \), from 0 up to \( \sqrt{z^2+1} \) due to the hyperboloid boundary.

An iterated integral efficiently processes each dimensional parameter step-by-step, turning a seemingly complex issue into more manageable pieces. In this exercise, each layer of computation, integrating over \( r \), \( \theta \), and \( z \), contributes to calculating the solid's mass, which inherently possesses the solid's volume characteristics scaled by density. The inclusion of a Jacobian factor, \( r \), accounts for the transformation from Cartesian to cylindrical coordinates, ensuring correct volume scaling.