Indefinite and Definite Integral Calculator (Numerical Approximation)
Understanding integrals is fundamental in calculus, with applications ranging from calculating areas and volumes to modeling physical phenomena. While an indefinite integral (also known as an antiderivative) represents a family of functions whose derivative is the original function, a definite integral calculates the net signed area between a function's graph and the x-axis over a specified interval.
A simple web calculator cannot symbolically solve indefinite integrals (e.g., find the antiderivative of sin(x) as -cos(x) + C). However, we can numerically approximate definite integrals, which provides a concrete numerical value. This calculator uses the Trapezoidal Rule to estimate the definite integral of a simple quadratic function f(x) = Ax² + Bx + C over a given interval.
How to Use This Calculator:
Enter the coefficients for your quadratic function, the lower and upper limits of integration, and the number of subintervals for the approximation. A higher number of subintervals generally leads to a more accurate approximation.
Result:
" + A + "x² + " + B + "x + " + C + " from " + a + " to " + b + " is: " + integralValue.toFixed(6) + "";
}
What is an Indefinite Integral?
An indefinite integral, or antiderivative, is the reverse process of differentiation. If you have a function f(x), its indefinite integral, denoted as ∫f(x)dx, is a new function F(x) such that the derivative of F(x) is f(x) (i.e., F'(x) = f(x)). Because the derivative of a constant is zero, indefinite integrals always include an arbitrary constant of integration, typically denoted as C. For example, the indefinite integral of 2x is x² + C.
What is a Definite Integral?
A definite integral, denoted as ∫abf(x)dx, calculates the net signed area between the curve of f(x) and the x-axis over a specific interval [a, b]. Unlike indefinite integrals, definite integrals yield a single numerical value, not a family of functions. The Fundamental Theorem of Calculus links these two concepts: if F(x) is an antiderivative of f(x), then ∫abf(x)dx = F(b) - F(a).
Numerical Integration: The Trapezoidal Rule
Since finding an exact antiderivative can be complex or impossible for some functions, numerical integration methods are used to approximate definite integrals. The Trapezoidal Rule is one such method. It approximates the area under the curve by dividing the interval [a, b] into smaller subintervals and forming trapezoids under the curve in each subinterval. The sum of the areas of these trapezoids gives an approximation of the total area.
The formula for the Trapezoidal Rule is:
∫abf(x)dx ≈ (h/2) * [f(a) + 2f(x₁) + 2f(x₂) + ... + 2f(xn-1) + f(b)]
where h = (b - a) / n is the width of each subinterval, and xᵢ = a + i*h are the points within the interval.
Examples:
Let's calculate the definite integral of f(x) = x² from 0 to 1 using 100 subintervals:
- Coefficient of x² (A): 1
- Coefficient of x (B): 0
- Constant Term (C): 0
- Lower Limit (a): 0
- Upper Limit (b): 1
- Number of Subintervals (n): 100
- Result: Approximately 0.333350 (The exact value is 1/3 or 0.333333…)
Now, let's try f(x) = 2x + 3 from -1 to 2 with 500 subintervals:
- Coefficient of x² (A): 0
- Coefficient of x (B): 2
- Constant Term (C): 3
- Lower Limit (a): -1
- Upper Limit (b): 2
- Number of Subintervals (n): 500
- Result: Approximately 12.000000 (The exact value is 12)