Left Riemann Sum
Right Riemann Sum
Midpoint Riemann Sum
Understanding the Reimann Sum Calculator
The Reimann Sum calculator is a powerful tool used in calculus to approximate the definite integral of a function. A definite integral represents the area under the curve of a function between two specified points on the x-axis. Since finding the exact area can be complex for many functions, Reimann sums provide a method to estimate this area by dividing it into a series of rectangles.
How it Works
The core idea is to approximate the area under the curve of a function, f(x), between an interval [a, b]. The calculator works by:
Defining the Interval: The user specifies the lower bound (a) and the upper bound (b) of the interval on the x-axis.
Partitioning the Interval: The interval [a, b] is divided into 'n' smaller subintervals of equal width. The width of each subinterval, denoted as Δx, is calculated as (b - a) / n.
Evaluating the Function at Specific Points: Within each subinterval, the function f(x) is evaluated at a specific point. The type of Reimann sum determines which point is chosen:
Left Reimann Sum: The function is evaluated at the left endpoint of each subinterval.
Right Reimann Sum: The function is evaluated at the right endpoint of each subinterval.
Midpoint Reimann Sum: The function is evaluated at the midpoint of each subinterval.
Calculating Rectangle Areas: For each subinterval, a rectangle is formed. The width of the rectangle is Δx, and its height is the value of the function f(x) at the chosen point within that subinterval. The area of each rectangle is therefore f(x_i) * Δx.
Summing the Areas: The Reimann sum is the total sum of the areas of all these rectangles. As the number of intervals (n) increases, the approximation becomes more accurate, converging towards the true value of the definite integral.
Formula
The general formula for a Reimann sum is:
Sum = Σ [f(x_i) * Δx] from i = 1 to n
Where:
Σ denotes summation.
n is the number of subintervals.
Δx = (b - a) / n is the width of each subinterval.
x_i is the chosen point within the i-th subinterval (left endpoint, right endpoint, or midpoint).
Use Cases
The Reimann sum is a fundamental concept with wide applications:
Approximating Areas: Estimating the area under curves that are difficult or impossible to integrate analytically.
Calculating Work: In physics, it can be used to calculate the work done by a variable force over a distance.
Finding Displacement: Estimating the displacement of an object given its velocity function.
Foundation for Integration: It forms the theoretical basis for the definite integral.
Example
Let's approximate the integral of f(x) = x^2 from a = 0 to b = 4 using n = 4 intervals and a Left Reimann Sum.
The exact integral is [x^3 / 3] from 0 to 4 = (4^3 / 3) - (0^3 / 3) = 64 / 3 ≈ 21.33. The Reimann sum provides an approximation that improves with more intervals.
function calculateReimannSum() {
var funcStr = document.getElementById("function").value;
var a = parseFloat(document.getElementById("lowerBound").value);
var b = parseFloat(document.getElementById("upperBound").value);
var n = parseInt(document.getElementById("numIntervals").value);
var type = document.getElementById("reimannType").value;
var resultDiv = document.getElementById("result");
// Clear previous results
resultDiv.innerHTML = "";
// Input validation
if (isNaN(a) || isNaN(b) || isNaN(n) || n = b) {
resultDiv.innerHTML = "Error: Lower bound (a) must be less than the upper bound (b).";
return;
}
// Define a function to evaluate the user's input function string
var f = function(x) {
try {
// Basic sanitization and replacement for common math functions
var sanitizedFuncStr = funcStr.replace(/Math\.pow/g, 'Math.pow')
.replace(/Math\.sin/g, 'Math.sin')
.replace(/Math\.cos/g, 'Math.cos')
.replace(/Math\.tan/g, 'Math.tan')
.replace(/Math\.sqrt/g, 'Math.sqrt')
.replace(/Math\.log/g, 'Math.log')
.replace(/Math\.exp/g, 'Math.exp');
// Using Function constructor for dynamic evaluation – BE CAREFUL with untrusted input in real applications
// For this calculator, we assume the input is from a trusted source or context.
var evalFunc = new Function('x', 'Math', 'return ' + sanitizedFuncStr);
return evalFunc(x, Math);
} catch (e) {
console.error("Error evaluating function:", e);
return NaN; // Indicate an error occurred
}
};
var deltaX = (b – a) / n;
var totalSum = 0;
for (var i = 0; i < n; i++) {
var x_i;
if (type === "left") {
x_i = a + i * deltaX;
} else if (type === "right") {
x_i = a + (i + 1) * deltaX;
} else { // midpoint
x_i = a + (i + 0.5) * deltaX;
}
var fx_i = f(x_i);
if (isNaN(fx_i)) {
resultDiv.innerHTML = "Error: Could not evaluate the function at a required point. Check your function input.";
return;
}
totalSum += fx_i * deltaX;
}
// Display the result
if (!isNaN(totalSum)) {
resultDiv.innerHTML = "Approximated Area: " + totalSum.toFixed(6);
} else {
resultDiv.innerHTML = "Error: Calculation resulted in an invalid number.";
}
}