Estimating Areas under Curves

Estimating the area under a curve involves determining how much "space" a particular section of the graph occupies. Though the shapes may not always be regular, certain strategies can help us approximate the area sufficiently.

Why Estimate Areas?

Estimating areas under curves is essential for problems where exact answers are either not necessary or impossible to obtain with basic algebra. This includes calculating distances, rates of change, or any situation where the total accumulation of a quantity over time is needed.

Methods for Estimating Area

Several approaches can be taken to approximate the area under a curve:

• Rectangular Approximation: Dividing the area into a series of rectangles, estimating the area of each, and summing them up.
  • So for this method you mark points on the graph and create rectangles that cover the area under the curve
• Trapezoidal Rule: Similar to the rectangular method, but uses trapezoids instead of rectangles, which can provide a better approximation by following the curve more closely.
  • Similar to the rectangle method you draw points on the line and jjoin these points using stright lines forming a trapezium shape. 
  • The area is then calculated by working out each trapezium area and summing all the trapezium areas

Steps to Estimate Area

1. Choose the Method: Decide whether to use rectangles, trapezoids, or another approach based on the curve and the level of accuracy required.
2. Divide the Interval: Break down the area under consideration into smaller sections (intervals).
3. Calculate Individual Areas: Determine the area of each section based on your chosen method.
4. Sum the Areas: Add up the areas of all sections to get an overall estimate of the total area.