Calculate the Area Under a Curve

Area Under a Curve Calculator (Quadratic Function)

function calculateAreaUnderCurve() { var coeffA = parseFloat(document.getElementById('coeffA').value); var coeffB = parseFloat(document.getElementById('coeffB').value); var coeffC = parseFloat(document.getElementById('coeffC').value); var lowerLimit = parseFloat(document.getElementById('lowerLimit').value); var upperLimit = parseFloat(document.getElementById('upperLimit').value); var numIntervals = parseInt(document.getElementById('numIntervals').value); if (isNaN(coeffA) || isNaN(coeffB) || isNaN(coeffC) || isNaN(lowerLimit) || isNaN(upperLimit) || isNaN(numIntervals) || numIntervals = upperLimit) { document.getElementById('areaResult').innerHTML = 'The lower limit must be less than the upper limit.'; return; } // Define the function f(x) = Ax^2 + Bx + C var f = function(x) { return coeffA * x * x + coeffB * x + coeffC; }; // Calculate h (width of each sub-interval) var h = (upperLimit – lowerLimit) / numIntervals; var sum = 0.5 * (f(lowerLimit) + f(upperLimit)); // Start with 0.5 * (f(a) + f(b)) // Apply the trapezoidal rule: sum f(xi) for i=1 to n-1 for (var i = 1; i < numIntervals; i++) { var x_i = lowerLimit + i * h; sum += f(x_i); } var approximateArea = h * sum; document.getElementById('areaResult').innerHTML = 'Approximate Area: ' + approximateArea.toFixed(6); }

Understanding the Area Under a Curve

The "area under a curve" is a fundamental concept in calculus, representing the definite integral of a function between two specified points on the x-axis. Geometrically, it's the area of the region bounded by the function's graph, the x-axis, and the vertical lines at the lower and upper limits.

Why is it Important?

Calculating the area under a curve has vast applications across various fields:

• Physics: The area under a velocity-time graph gives displacement. The area under a force-distance graph gives work done.
• Engineering: Used in structural analysis, fluid dynamics, and signal processing.
• Economics: Can represent total cost, total revenue, or consumer/producer surplus.
• Statistics and Probability: The area under a probability density function represents the probability of an event occurring within a certain range.
• Biology: Used in pharmacokinetics to calculate drug exposure (Area Under the Curve – AUC).

How is it Calculated? Numerical Integration

While analytical methods (antiderivatives) are precise for many functions, some functions are difficult or impossible to integrate analytically. In such cases, numerical integration techniques are employed to approximate the area. This calculator uses the Trapezoidal Rule, a common numerical method.

The Trapezoidal Rule Explained

The trapezoidal rule approximates the area under a curve by dividing the region into a series of trapezoids instead of rectangles (as in Riemann sums). For each small interval along the x-axis, a trapezoid is formed by connecting the function's values at the interval's endpoints. The area of each trapezoid is calculated, and then all these areas are summed up to get the total approximate area.

The more sub-intervals (n) you use, the narrower each trapezoid becomes, and the closer the approximation gets to the true area under the curve. This calculator allows you to define a quadratic function of the form f(x) = Ax² + Bx + C and specify the integration limits (a and b) and the number of sub-intervals (n) for the approximation.

Example Calculation

Let's calculate the area under the curve f(x) = x² from x = 0 to x = 2.

• Coefficient A: 1
• Coefficient B: 0
• Coefficient C: 0
• Lower Limit (a): 0
• Upper Limit (b): 2
• Number of Sub-intervals (n): 1000 (a good number for reasonable accuracy)

Using the calculator with these inputs, you would find an approximate area very close to 2.666667. The exact analytical solution for the integral of x² from 0 to 2 is [x³/3] from 0 to 2, which evaluates to (2³/3) – (0³/3) = 8/3 ≈ 2.666667.