Integral Calculator

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Integral Calculator
Definite Integral (f(x) = Ax² + Bx + C)
Result:
Integral Value = —
function calculateIntegral(){var a=parseFloat(document.getElementById('coeffA').value);var b=parseFloat(document.getElementById('coeffB').value);var c=parseFloat(document.getElementById('coeffC').value);var lower=parseFloat(document.getElementById('lowerBound').value);var upper=parseFloat(document.getElementById('upperBound').value);var showSteps=document.getElementById('steps').checked;if(isNaN(a)||isNaN(b)||isNaN(c)||isNaN(lower)||isNaN(upper)){alert('Please enter valid numerical coefficients and limits.');return;}var F_upper=(a/3)*Math.pow(upper,3)+(b/2)*Math.pow(upper,2)+(c*upper);var F_lower=(a/3)*Math.pow(lower,3)+(b/2)*Math.pow(lower,2)+(c*lower);var result=F_upper-F_lower;document.getElementById('resultValue').innerHTML=result.toFixed(4);if(showSteps){var stepText="Antiderivative F(x) = ("+(a/3).toFixed(2)+")x³ + ("+(b/2).toFixed(2)+")x² + ("+c.toFixed(2)+")x
";stepText+="Calculate F("+upper+") – F("+lower+")";document.getElementById('stepBox').innerHTML=stepText;document.getElementById('stepBox').style.display='block';}else{document.getElementById('stepBox').style.display='none';}}

Calculator Use

The integral calculator is a specialized tool designed to solve definite integrals for polynomial functions of the form f(x) = Ax² + Bx + C. In calculus, integration is the process of finding the area under a curve, representing the accumulation of quantities. This calculator is particularly useful for students and professionals who need quick, accurate results for area calculations without manual derivation.

To use this calculator, simply enter the coefficients for your quadratic or linear function and set the interval limits. The tool applies the Fundamental Theorem of Calculus to provide the exact numerical result.

Coefficient A, B, and C
These represent the multipliers for the x², x, and constant terms of your function. For a simple line like f(x) = 2x + 5, you would set A=0, B=2, and C=5.
Lower Limit (a)
The starting point on the x-axis for the integration interval.
Upper Limit (b)
The ending point on the x-axis for the integration interval.

How It Works

The integral calculator utilizes the Power Rule for integration. This rule states that the integral of x to the power of n is (x^(n+1))/(n+1). For a polynomial function, we integrate each term individually. The formula for a definite integral from a to b is expressed as:

∫[a to b] (Ax² + Bx + C) dx = [ (A/3)x³ + (B/2)x² + Cx ] evaluated from a to b

  • Antiderivative: First, we find the general function F(x) whose derivative is f(x).
  • Evaluation: We plug the upper limit (b) into F(x).
  • Subtraction: We plug the lower limit (a) into F(x) and subtract that value from F(b).
  • Result: The final number represents the "signed" area between the function and the x-axis.

Calculation Example

Example: Find the area under the curve 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. Calculate F(3): (3)³ + 2(3)² + 2(3) = 27 + 18 + 6 = 51.
  4. Calculate F(1): (1)³ + 2(1)² + 2(1) = 1 + 2 + 2 = 5.
  5. Subtract: 51 – 5 = 46.
  6. Result = 46.0000

Common Questions

What is the difference between definite and indefinite integrals?

An indefinite integral represents a family of functions (the antiderivative) and includes a constant "C" because the derivative of any constant is zero. A definite integral, which this integral calculator computes, has specific upper and lower limits and results in a single numerical value representing the accumulation over that interval.

Can an integral result be negative?

Yes. While we often think of integrals as area, the integral measures "signed area." If the function lies below the x-axis within the chosen limits, the resulting integral value will be negative. This is common in physics applications, such as calculating displacement where direction matters.

Why is the Power Rule important in calculus?

The Power Rule is the fundamental building block for integration. It allows us to handle polynomials, which are the most common functions in introductory physics and engineering. Without the power rule, calculating the work done by a variable force or the center of mass of an object would be significantly more complex.

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