Problem 33 - Calculus: Early Transcendentals

Question

Use calculus to find the area of the triangle with the given vertices. \( (0 , 0) \) , \( (3 , 1) \) , \( (1 , 2) \)

Step-by-Step Solution

Answer

The area of the triangle is 2.5 square units.

Step 1: Understand the Problem

We need to find the area of the triangle formed by the three points: \((0, 0)\), \((3, 1)\), and \((1, 2)\). We can use the formula for the area of a triangle given by three coordinates calculated using the determinative approach.

Step 2: Set Up the Determinant Formula

The area \( A \) of a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) is given by the formula:\[ A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]

Step 3: Substitute Coordinates into the Formula

Let's substitute the given vertices into the formula. For our coordinates:- \((x_1, y_1) = (0, 0)\)- \((x_2, y_2) = (3, 1)\)- \((x_3, y_3) = (1, 2)\)We substitute them into our formula:\[ A = \frac{1}{2} \left| 0(1-2) + 3(2-0) + 1(0-1) \right| \]

Step 4: Simplify the Expression

Let's simplify the expression inside the absolute value:\[ A = \frac{1}{2} \left| 0 + 6 - 1 \right| = \frac{1}{2} \left| 5 \right| \]

Step 5: Calculate the Area

Now, calculate the area:\[ A = \frac{1}{2} \times 5 = 2.5 \]So, the area of the triangle is 2.5 square units.

Key Concepts

Determinant Method

The determinant method is a straightforward and powerful way to calculate the area of a triangle using the coordinates of its vertices. This method takes advantage of a mathematical formula that applies the determinant of a matrix conceptualized from the points.
To apply this method, consider a triangle with vertices at points

• \((x_1, y_1)\),
• \((x_2, y_2)\), and
• \((x_3, y_3)\).

The area \( A \) can be found using:\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]This formula results from calculating the determinant of a 2x2 matrix derived from the vertex coordinates, succinctly capturing the signed area of the parallelogram formed by the vectors from an origin, which is the basis for deriving the triangle area. Using absolute values ensures that the area is non-negative, as area cannot be negative.

Calculus Application

In the context of finding the area of geometric shapes, calculus provides various tools that can be instrumental. Though the determinant method is primarily used in this specific problem, calculus concepts, such as integration, underpin how we think about areas under curves and the space enclosed by boundaries.
Imagine plotting the points

• \((0, 0)\),
• \((3, 1)\), and
• \((1, 2)\)

on a graph. If you were to connect these points with lines, you would form the borders of the triangle. Calculus, specifically integration, helps us in other scenarios by summing up infinitesimally small sections of space to calculate total areas. Although direct integration isn't applied here, understanding the way calculus fits into geometric problems deepens comprehension.
It emphasizes that even seemingly simple geometric problems have deep connections to the broad field of calculus, linking discrete coordinate geometry with continuous curves and surfaces.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to use algebra to solve geometric problems. It introduces a system where geometric shapes are represented through mathematical equations, using coordinates on the Cartesian plane.
The vertices of a triangle

• \((0, 0)\),
• \((3, 1)\), and
• \((1, 2)\)

help form the basis of a problem that is fundamentally geometric, yet is approached with algebraic methods.
Calculating the area of a triangle using the determinant method is an example of how coordinate geometry eases the process. The vertices provide exact points, ensuring precision in calculations that would otherwise be cumbersome using only traditional geometric means.
Moreover, this practice allows an abstraction in solving geometric problems, where similar principles applied here can extend to more complex figures or even three dimensions. Thus, coordinate geometry is not only confined to triangles but becomes a universal approach to understanding geometric spaces.