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Write geometric area as a function of x

Use the geometry formula, then remove extra variables

Start with the ordinary area formula. Use the given graph, perimeter or other constraint to express every remaining length in terms of $x$ . State the physical domain when $x$ represents a positive length.

Problem

Example 1. A right triangle has vertices $(0, 0)$ , $(0, y)$ and $(x, y)$ , with $(x, y)$ on $y = x^{8}$ and $x > 0$ . Express its area as a function of $x$ . Example 2. A wire of length $5 x$ is bent into a circle, where $x > 0$ . Express the circle's circumference and area as functions of $x$ .

Answer

Triangle:

A (x) = \frac{1}{2} x^{9}

. Circle:

C (x) = 5 x

and

A (x) = \frac{25 x ^{2}}{4 π}

Both answers use x > 0 because the dimensions are physical lengths.

Step-by-step solution

Example 1: triangle on a curve

The horizontal leg has length $x$ and the vertical leg has length $y$ , so $A = x y /2$ . Since the upper-right vertex is on $y = x^{8}$ , substitute:

A (x) = \frac{1}{2} x \cdot x^{8} = \frac{1}{2} x^{9} .

Example 2: circle from a wire

The full wire becomes the boundary, so $C (x) = 5 x$ . From $C = 2 π r$ ,

r = \frac{5 x}{2 π} .

Substitute into $A = π r^{2}$ :

A (x) = π (\frac{5 x}{2 π})^{2} = \frac{25 x ^{2}}{4 π} .

The common method is to substitute the constraint before simplifying. The formulas apply for positive $x$ .

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