Area Under the Curve Calculator
Calculate the definite integral of a function $f(x) = Ax^2 + Bx + C$ using the Trapezoidal Rule.
Calculation Result
0.00
Understanding the Area Under a Graph
The area under a graph (or the definite integral) represents the cumulative total of a variable over an interval. In physics, this might represent displacement (area under a velocity-time graph) or work done (area under a force-displacement graph). In statistics, it represents probability density.
The Trapezoidal Rule Formula
This calculator uses the Trapezoidal Rule, a numerical integration method that approximates the region under the graph of a function as a series of trapezoids. The formula is:
Area ≈ (Δx / 2) * [f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(xₙ₋₁) + f(xₙ)]
Where:
- Δx = (b – a) / n
- a = Lower limit of integration
- b = Upper limit of integration
- n = Number of sub-intervals
Real-World Example
Imagine a vehicle accelerating where the velocity function is v(t) = 2t² + 3t. To find the total distance traveled between 0 and 4 seconds:
- Set Lower Limit (a) to 0.
- Set Upper Limit (b) to 4.
- Set Coefficient A to 2, B to 3, and C to 0.
- Increase Intervals (n) (e.g., 100) for higher precision.
- The result provides the total distance in units (e.g., meters).