Area Under Curve Calculator
Use this calculator to approximate the area under a user-defined function using the Trapezoidal Rule. This method is a numerical integration technique that approximates the definite integral of a function by dividing the area under the curve into a series of trapezoids.
Understanding 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 vertical lines at the lower and upper bounds of the interval.
Why is it Important?
- Physics: The area under a velocity-time graph gives displacement; the area under a force-distance graph gives work done.
- Engineering: Calculating volumes, centroids, and moments of inertia.
- Economics: Consumer and producer surplus, total cost, and total revenue can be found by integrating relevant functions.
- Statistics and Probability: The area under a probability density function represents the probability of an event occurring within a certain range.
- Data Analysis: Estimating cumulative effects or totals from rate data.
The Trapezoidal Rule
When an exact analytical solution for an integral is difficult or impossible to find, or when dealing with discrete data points, numerical integration methods are used. The Trapezoidal Rule is one such method. It approximates the area under the curve by dividing the interval into smaller sub-intervals and treating the area above each sub-interval as a trapezoid.
For each sub-interval, the area of the trapezoid is calculated as: (f(x₀) + f(x₁)) / 2 * h, where f(x₀) and f(x₁) are the function values at the start and end of the sub-interval, and h is the width of the sub-interval. The total area is the sum of the areas of all these trapezoids.
The accuracy of the Trapezoidal Rule generally increases with the number of segments (n) used, as the approximation of the curve by straight lines becomes finer.
How to Use the Calculator
- Function f(x): Enter your mathematical function. Use 'x' as the variable. For mathematical constants and functions, use the
Math. prefix (e.g., Math.sin(x), Math.PI, Math.exp(x)).
- Lower Bound (a): Enter the starting x-value of the interval.
- Upper Bound (b): Enter the ending x-value of the interval. This value must be greater than the lower bound.
- Number of Segments (n): Specify how many trapezoids the interval should be divided into. A higher number generally yields a more accurate result but takes slightly longer to compute.
- Click "Calculate Area" to see the approximated area under your curve.
Example Calculation
Let's calculate the area under the curve f(x) = x*x from x = 0 to x = 2.
- Function f(x):
x*x
- Lower Bound (a):
0
- Upper Bound (b):
2
- Number of Segments (n):
100
The exact analytical integral of x*x from 0 to 2 is [x^3 / 3] evaluated from 0 to 2, which is (2^3 / 3) - (0^3 / 3) = 8 / 3 = 2.666.... Using the calculator with 100 segments will provide a very close approximation to this value.
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function evaluateFunction(funcStr, xVal) {
try {
// Replace common math functions with Math. prefix for safety and functionality
var processedFuncStr = funcStr
.replace(/sin\(/g, 'Math.sin(')
.replace(/cos\(/g, 'Math.cos(')
.replace(/tan\(/g, 'Math.tan(')
.replace(/log\(/g, 'Math.log(') // Natural log
.replace(/log10\(/g, 'Math.log10(') // Base 10 log
.replace(/exp\(/g, 'Math.exp(')
.replace(/sqrt\(/g, 'Math.sqrt(')
.replace(/abs\(/g, 'Math.abs(')
.replace(/round\(/g, 'Math.round(')
.replace(/ceil\(/g, 'Math.ceil(')
.replace(/floor\(/g, 'Math.floor(')
.replace(/PI/g, 'Math.PI')
.replace(/E/g, 'Math.E')
.replace(/\^/g, '**'); // Replace ^ with ** for power operator
// Create a function dynamically to evaluate the string
// This approach is generally safer than direct eval() as it scopes 'x'
var func = new Function('x', 'return ' + processedFuncStr + ';');
return func(xVal);
} catch (e) {
console.error("Error evaluating function:", e);
return NaN; // Indicate an error
}
}
function calculateAreaUnderCurve() {
var functionInput = document.getElementById("functionInput").value;
var lowerBound = parseFloat(document.getElementById("lowerBound").value);
var upperBound = parseFloat(document.getElementById("upperBound").value);
var numSegments = parseInt(document.getElementById("numSegments").value);
var resultDiv = document.getElementById("calculationResult");
if (isNaN(lowerBound) || isNaN(upperBound) || isNaN(numSegments)) {
resultDiv.innerHTML = "
Please enter valid numbers for all fields.";
return;
}
if (upperBound <= lowerBound) {
resultDiv.innerHTML = "
Upper Bound must be greater than Lower Bound.";
return;
}
if (numSegments <= 0) {
resultDiv.innerHTML = "
Number of Segments must be a positive integer.";
return;
}
var h = (upperBound – lowerBound) / numSegments;
var area = 0;
// Trapezoidal Rule implementation
for (var i = 0; i < numSegments; i++) {
var x0 = lowerBound + i * h;
var x1 = lowerBound + (i + 1) * h;
var y0 = evaluateFunction(functionInput, x0);
var y1 = evaluateFunction(functionInput, x1);
if (isNaN(y0) || isNaN(y1)) {
resultDiv.innerHTML = "
Error: Invalid function or syntax. Please check your function input.";
return;
}
area += (y0 + y1) / 2 * h;
}
resultDiv.innerHTML = "Approximated Area:
" + area.toFixed(6) + "";
}