Area Under a Curve Calculator (Quadratic Function)
This calculator estimates the area under a quadratic curve defined by f(x) = Ax² + Bx + C over a specified interval using the Trapezoidal Rule. The Trapezoidal Rule is a numerical integration method that approximates the area by dividing the region under the curve into a series of trapezoids.
Calculated Area:
Understanding the Area Under a Curve
The "area under a curve" is a fundamental concept in calculus, representing the definite integral of a function over a given interval. Geometrically, it's the area of the region bounded by the function's graph, the x-axis, and the vertical lines at the start and end of the interval. This concept has vast applications across various fields:
- Physics: The area under a velocity-time graph gives displacement; under a force-distance graph gives work done.
- Engineering: Calculating volumes, centroids, and moments of inertia.
- Statistics: The area under a probability density function represents the probability of an event occurring within a certain range.
- Economics: Consumer and producer surplus can be found by calculating areas under demand and supply curves.
The Trapezoidal Rule for Numerical Integration
While analytical integration provides exact solutions for many functions, some functions are difficult or impossible to integrate analytically. In such cases, or when dealing with discrete data points, numerical integration methods are used to approximate the area. The Trapezoidal Rule is one of the simplest and most commonly used methods.
The core idea of the Trapezoidal Rule is to approximate the region under the curve as a series of trapezoids rather than rectangles (as in Riemann sums). For a function f(x) over an interval [a, b], divided into n equal subintervals, each of width h = (b - a) / n, the area is approximated by the sum of the areas of these trapezoids.
The formula for the Trapezoidal Rule is:
Area ≈ (h / 2) * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]
Where:
his the width of each subinterval.x₀, x₁, ..., xₙare the x-coordinates of the endpoints of the subintervals, withx₀ = aandxₙ = b.f(xᵢ)is the function's value at each x-coordinate.
As the number of subintervals (n) increases, the approximation generally becomes more accurate, approaching the true value of the definite integral.
How to Use This Calculator
This calculator is designed for quadratic functions of the form f(x) = Ax² + Bx + C. Follow these steps:
- Enter Coefficients A, B, and C: Input the numerical values for the coefficients of your quadratic function. For example, for
f(x) = x², you would enter A=1, B=0, C=0. Forf(x) = 2x² + 3x - 5, you would enter A=2, B=3, C=-5. - Define the Interval: Enter the 'Start of Interval (a)' and 'End of Interval (b)' to specify the range over which you want to calculate the area.
- Set Number of Subintervals (n): Choose a positive integer for the number of subintervals. A higher number generally leads to a more accurate approximation but takes slightly longer to compute. A value of 100 or more is often a good starting point for reasonable accuracy.
- Click "Calculate Area": The calculator will then display the estimated area under the curve for your specified function and interval.
Example Calculation
Let's calculate the area under the curve f(x) = x² from x = 0 to x = 2 using 100 subintervals.
- Coefficient A: 1
- Coefficient B: 0
- Coefficient C: 0
- Start of Interval (a): 0
- End of Interval (b): 2
- Number of Subintervals (n): 100
Using the Trapezoidal Rule, the calculator would perform the following steps:
- Calculate
h = (2 - 0) / 100 = 0.02. - Evaluate
f(x) = x²atx=0andx=2:f(0) = 0,f(2) = 4. - Sum
f(x)for intermediate points, multiplying by 2. - Apply the Trapezoidal Rule formula.
The exact integral of x² from 0 to 2 is [x³/3] from 0 to 2, which is (2³/3) - (0³/3) = 8/3 ≈ 2.666667. With 100 subintervals, the Trapezoidal Rule will provide a very close approximation to this value.