Area of a Triangle
The Setup
Consider the first quadrant of the Cartesian plane. Demark a point on the positive y-axis as (0,b), this will serve as the y-intercept for our linear function. Likewise, mark a point on the positive x-axis as (a,0) which will serve as our x-coordinate.
Draw a line through these two points. You should see a right triangle formed by the line and the positive coordinate axes.
The Triangular Area
We wish to determine what is the bounded area between the linear function and the two coordinate axes. Since we know the area of a rectangle is the product of the base and its height, let's estimate the bounded area with three rectangles whose left-hand corner will intersect the line.
The base of each rectangle will be equally divided by breaking up the length of the segment OA into the same number of rectangles we need to determine the bounded area.
Compute the Area
The three rectangles shown above is an overestimate of the area, as the overlap shows. To more accurately map the bounded area we would need to increase the number of rectangles we are using to estimate the bounded area.
In fact, in the limit of infinitely many rectangles, we obtain the exact value of the bounded area.
Set the stage, the ordered pairs of the triangles' vertices are O=(0,0), A=(a,0), and B=(0,b).
1. Determine the equation of the line through points A and B.
2. The limits of integration are the x-coordinates of the points O and A.
3. Integrate the linear function over the limits of integration and simplify to arrive at a familiar result.