Problem 49 - Thomas Calculus

Question

A surface \(S\) lies on the plane \(2 x+3 y+6 z=12\) directly above the rectangle in the \(x y\) -plane with vertices \((0,0),(1,0),\) \((0,2),\) and \((1,2) .\) If the density at a point \((x, y, z)\) on \(S\) is given by \(\delta(x, y, z)=4 x y+6 z \mathrm{mg} / \mathrm{cm}^{2},\) find the total mass of \(S .\)

Step-by-Step Solution

Answer

The total mass of the surface is 20 mg/cm².

Step 1: Describe the Plane Equation

The plane equation given is \(2x + 3y + 6z = 12\). We can express \(z\) as a function of \(x\) and \(y\) by solving for \(z\). Rearrange the equation to get \(z = \frac{12 - 2x - 3y}{6}\), which simplifies to \(z = 2 - \frac{x}{3} - \frac{y}{2}\).

Step 2: Define the Region R

The region \(R\) over which the surface lies is a rectangle in the \(xy\)-plane, with vertices at \((0,0), (1,0), (0,2),\) and \((1,2)\). This means \(0 \leq x \leq 1\) and \(0 \leq y \leq 2\).

Step 3: Express Density Function

The density function given is \(\delta(x, y, z) = 4xy + 6z\mathrm{\ mg/cm^2}\). Substitute \(z = 2 - \frac{x}{3} - \frac{y}{2}\) into the density function to express \(\delta\) in terms of \(x\) and \(y\): \(\delta(x, y) = 4xy + 6\left(2 - \frac{x}{3} - \frac{y}{2}\right)\).

Step 4: Simplify the Density Function

Simplify \(\delta(x, y) = 4xy + 12 - 2x - 3y\). This is the density function to be integrated over the rectangle \(0 \leq x \leq 1\) and \(0 \leq y \leq 2\).

Step 5: Set Up the Mass Integral

The mass \(M\) of the surface \(S\) is given by the double integral \(M = \int_0^1 \int_0^2 (4xy + 12 - 2x - 3y) \, dy \, dx\).

Step 6: Integrate with Respect to y

First, integrate the expression \(4xy + 12 - 2x - 3y\) with respect to \(y\): \(\int_0^2 (4xy + 12 - 2x - 3y) \, dy = [2xy^2 + 12y - 2xy - \frac{3}{2}y^2]_0^2 \).Evaluating gives: \(8x + 24 - 4x - 6 = 4x + 18\).

Step 7: Integrate with Respect to x

Now integrate \(4x + 18\) with respect to \(x\):\(\int_0^1 (4x + 18) \, dx = [2x^2 + 18x]_0^1 \).Evaluating gives: \(2 + 18 \cdot 1 = 20\).

Step 8: Conclude with the Total Mass

After evaluating both integrals, the total mass \(M\) of the surface \(S\) is \(20\ \mathrm{mg/cm^2}\).

Key Concepts

Surface Integration

Surface integration is a technique used to find various properties of a surface, like mass or area. In the context of this exercise, surface integration helps calculate the total mass of a given surface. The integration is performed by considering the contributions from both the shape of the surface and the density function applied to it.

Surface integration requires two main steps:

• Setting up the integral using the parameterized surface.
• Integrating the density function over the defined region on the surface.

Here, the surface is derived from the plane equation, and the rectangle defines the domain over which the surface is laid out. The core aim is to evaluate the integral that represents the mass, taking into account the varying density across the surface.

Density Function

The density function expresses how mass is distributed over a particular surface. It tells us how much mass is concentrated in different parts of the surface. In this exercise, the density function is given by \(\delta(x, y, z) = 4xy + 6z \, \text{mg/cm}^2 \).

This function includes both variables from the plane equation and is expressed in terms of the plane's coordinates.

• The term \(4xy\) describes the density contribution based on the plane's position along the x and y directions.
• The term \(6z\) accounts for the density along the z-axis, which changes as per the plane equation solved earlier.

By substituting the expression for \(z\) from the plane equation, the function gets simplified to depend only on x and y. This modified density function is crucial for calculating the overall mass through integration.

Multiple Integrals

Multiple integrals extend the concept of single-variable integration to functions of two or more variables. In this exercise, to find the mass of the surface, a double integral is used because the density function is dependent on both x and y.

The integration process involves two main steps:

• First, integrating with respect to y, which considers changes along the vertical direction within the rectangle defined in the xy-plane.
• Then, integrating the resulting expression with respect to x, which captures changes along the horizontal direction.

Each of these integrals simplifies the function step by step until the total surface mass is obtained by evaluating the bounds of the integral. Handling multiple integrals effectively requires understanding how to decompose these into manageable parts, handled through sequential integration.

Plane Equation

A plane equation represents a flat, two-dimensional surface in three-dimensional space. In this exercise, the plane given by the equation \(2x + 3y + 6z = 12\) describes the surface on which mass is distributed.

The task initially requires solving for \(z\) in terms of \(x\) and \(y\). By rearranging the equation, we find

• \(z = 2 - \frac{x}{3} - \frac{y}{2}\)

This expression is crucial because it transforms the plane into a form that's handy for substitution into the density function later.
Understanding a plane equation involves recognizing its components: the coefficients of x, y, and z, which influence the slope and orientation. By converting \(z\) to a function of \(x\) and \(y\), you prepare the equation for integration within the density function to find the mass connected to the surface.