Problem 35 - Calculus Volume 3

Question

Find the domain of the following functions. $$f(x, y, z)=\frac{1}{\sqrt{36-4 x^{2}-9 y^{2}-z^{2}}}$$

Step-by-Step Solution

Answer

Domain: \(4x^2 + 9y^2 + z^2 < 36\).

Step 1: Identify the Domain Constraints

To find the domain of the function, identify the constraints that must be satisfied. The expression inside the square root in the denominator must be positive to ensure the square root is defined and the fraction's denominator is not zero. Therefore, solve the inequality: \(36 - 4x^2 - 9y^2 - z^2 > 0\).

Step 2: Write the Inequality

Rewrite the inequality \(36 - 4x^2 - 9y^2 - z^2 > 0\) as \(4x^2 + 9y^2 + z^2 < 36\). This shows the condition under which \(f(x, y, z)\) is defined.

Step 3: Determine the Domain Region

The inequality \(4x^2 + 9y^2 + z^2 < 36\) describes an ellipsoid in three-dimensional space. The inequality means that the point \((x, y, z)\) must be inside the ellipsoid centered at the origin with semi-axis lengths 3, 2, and 6 for the x, y, and z axes, respectively.

Step 4: Express the Domain

The domain of the function \(f(x, y, z)\) is the set of all points \((x, y, z)\) that satisfy the inequality \(4x^2 + 9y^2 + z^2 < 36\).

Key Concepts

Ellipsoid

An ellipsoid is a three-dimensional geometric shape that can be thought of as a stretched or compressed sphere. Unlike a perfect sphere, an ellipsoid has three distinct semi-axes.

• These axes have different lengths, which define the shape's proportions.
• In our exercise, the ellipsoid is defined by the inequality \(4x^2 + 9y^2 + z^2 < 36\).
• This inequality represents an ellipsoid centered at the origin \((0,0,0)\).

The semi-axis lengths relate to the coefficients of \(x^2\), \(y^2\), and \(z^2\) in the inequality. Here, the coefficients are 4, 9, and 1 for \(x\), \(y\), and \(z\) respectively. This means the semi-axis lengths are \(\frac{6}{2} = 3\), \(\frac{6}{3} = 2\), and \(6\) for the x-axis, y-axis, and z-axis.

Domain Constraints

Domain constraints refer to the limitations or conditions under which a function is defined. These constraints ensure that the function does not reach undefined or indeterminate forms.

• For a function with a square root, the expression under the square root must always be positive.
• If the square root is in the denominator, it should never be zero to avoid undefined behavior.

In our function, \(f(x, y, z)=\frac{1}{\sqrt{36-4x^{2}-9y^{2}-z^{2}}} \), the constraint is that \(36-4x^{2}-9y^{2}-z^{2}\) must be positive. Therefore, the domain is the set of points \((x, y, z)\) that satisfy \(4x^2 + 9y^2 + z^2 < 36\).

Three-Dimensional Space

Three-dimensional space is the environment where objects have depth, width, and height. It is symbolized by three coordinate axes:

• The x-axis (horizontal direction)
• The y-axis (vertical direction)
• The z-axis (depth)

A point in this space is described by a triplet \((x, y, z)\). Most functions involving multiple variables are examined in this space.In our function's context, the domain, represented by the inequality \(4x^2 + 9y^2 + z^2 < 36\), describes a region of space inside an ellipsoid. This region is a solid shape where any point \((x, y, z)\) inside it satisfies the given inequality, defining where \(f(x, y, z)\) is valid in the three-dimensional space.