Integral Calculator

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Integral Calculator
Definite Integral (Polynomial)
Function: f(x) = Ax² + Bx + C
Result:
function calculateIntegral(){var a=parseFloat(document.getElementById('coeffA').value);var b=parseFloat(document.getElementById('coeffB').value);var c=parseFloat(document.getElementById('coeffC').value);var low=parseFloat(document.getElementById('lowerBound').value);var up=parseFloat(document.getElementById('upperBound').value);var steps=document.getElementById('showSteps').checked;if(isNaN(a)||isNaN(b)||isNaN(c)||isNaN(low)||isNaN(up)){alert('Please enter valid numerical values.');return;}var antiderivLow=(a*Math.pow(low,3)/3)+(b*Math.pow(low,2)/2)+(c*low);var antiderivUp=(a*Math.pow(up,3)/3)+(b*Math.pow(up,2)/2)+(c*up);var result=antiderivUp-antiderivLow;document.getElementById('finalResult').innerHTML='Area = '+result.toLocaleString(undefined,{minimumFractionDigits:2,maximumFractionDigits:4});if(steps){var stepHtml='Step 1: Find the Antiderivative F(x)
';stepHtml+='F(x) = ('+a+'/3)x³ + ('+b+'/2)x² + '+c+'x

';stepHtml+='Step 2: Evaluate at Upper Limit (b = '+up+')
';stepHtml+='F('+up+') = '+antiderivUp.toFixed(4)+'

';stepHtml+='Step 3: Evaluate at Lower Limit (a = '+low+')
';stepHtml+='F('+low+') = '+antiderivLow.toFixed(4)+'

';stepHtml+='Step 4: Subtract F(b) – F(a)
';stepHtml+='Result = '+antiderivUp.toFixed(4)+' – '+antiderivLow.toFixed(4)+' = '+result.toFixed(4);document.getElementById('stepDetails').innerHTML=stepHtml;document.getElementById('stepDetails').style.display='block';}else{document.getElementById('stepDetails').style.display='none';}document.getElementById('answer').style.display='block';}

Calculator Use

The integral calculator is a specialized tool designed to solve definite integrals for polynomial functions. By entering the coefficients of a quadratic or linear equation and specifying the interval limits, users can instantly determine the area under the curve. This tool is essential for students, engineers, and mathematicians who need quick verification of calculus problems without manual integration.

This calculator focuses on the Power Rule of integration, which is the foundation of introductory calculus. Whether you are calculating physical work, displacement, or geometric areas, the integral calculator provides precise results and step-by-step logic to aid learning.

Coefficient A (x²)
The multiplier for the squared term. Set to 0 if calculating a linear function.
Coefficient B (x)
The multiplier for the linear term.
Constant C
The numerical value added at the end of the function.
Lower and Upper Limits
The horizontal range (x-axis) over which the integral is calculated.

How It Works

Integration is the reverse process of differentiation. The integral calculator uses the Fundamental Theorem of Calculus to evaluate definite integrals. For a function f(x), the definite integral from a to b is calculated as:

∫[a to b] f(x) dx = F(b) – F(a)

  • f(x): The original function to be integrated.
  • F(x): The antiderivative of the function.
  • a: The lower limit of integration.
  • b: The upper limit of integration.

The Power Rule used by the integral calculator states that for any term xⁿ, the antiderivative is (xⁿ⁺¹)/(n+1). Our calculator applies this rule to each term of the polynomial separately.

Calculation Example

Example: Calculate the area under the curve for f(x) = 3x² + 4x + 2 between x = 1 and x = 3.

Step-by-step solution:

  1. Identify coefficients: A=3, B=4, C=2. Limits: a=1, b=3.
  2. Find the antiderivative: F(x) = (3/3)x³ + (4/2)x² + 2x = x³ + 2x² + 2x.
  3. Evaluate F(3): (3)³ + 2(3)² + 2(3) = 27 + 18 + 6 = 51.
  4. Evaluate F(1): (1)³ + 2(1)² + 2(1) = 1 + 2 + 2 = 5.
  5. Subtract: 51 – 5 = 46.
  6. Result = 46.

Common Questions

What is a definite integral?

A definite integral represents the signed area between the graph of a function and the x-axis over a specific interval. Unlike indefinite integrals, definite integrals result in a specific numerical value rather than a general formula with a constant 'C'.

Can an integral be negative?

Yes. If the function lies below the x-axis for the given interval, the integral calculator will yield a negative result. This indicates "negative area" in a mathematical context, though in physical applications like distance, you might take the absolute value.

What is the Fundamental Theorem of Calculus?

It is a theorem that links the concept of differentiating a function with the concept of integrating a function. It provides the computational method for evaluating definite integrals by using antiderivatives, which is exactly how our integral calculator operates.

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