Polynomial Definite Integral Calculator
This calculator helps you find the definite integral of a polynomial function of the form f(x) = Ax³ + Bx² + Cx + D over a specified interval [a, b]. While it calculates a definite integral, it relies on finding the indefinite integral (antiderivative) first, demonstrating a core concept of calculus.
Understanding Indefinite and Definite Integrals
In calculus, integration is the inverse operation of differentiation. It's a fundamental concept with wide applications in science, engineering, and economics.
Indefinite Integrals (Antiderivatives)
An indefinite integral, also known as an antiderivative, is a function F(x) whose derivative is the original function f(x). If F'(x) = f(x), then the indefinite integral of f(x) is written as ∫f(x) dx = F(x) + C. The + C represents the "constant of integration" because the derivative of any constant is zero. This means there's a whole family of functions that are antiderivatives of f(x), differing only by a constant.
For a polynomial function like f(x) = Ax³ + Bx² + Cx + D, the power rule for integration states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1). Applying this rule term by term, the indefinite integral would be:
F(x) = (A/4)x⁴ + (B/3)x³ + (C/2)x² + Dx + C_integration
Definite Integrals
A definite integral, on the other hand, calculates the net signed area between a function's graph and the x-axis over a specific interval [a, b]. It's represented as ∫_a^b f(x) dx. Unlike indefinite integrals, definite integrals evaluate to a single numerical value, not a family of functions.
The Fundamental Theorem of Calculus
The connection between indefinite and definite integrals is established by the Fundamental Theorem of Calculus. It states that if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b is given by:
∫_a^b f(x) dx = F(b) - F(a)
This theorem is incredibly powerful because it allows us to calculate definite integrals precisely by finding an antiderivative and evaluating it at the limits of integration, rather than relying solely on numerical approximations.
How This Calculator Works
This calculator takes the coefficients for a cubic polynomial f(x) = Ax³ + Bx² + Cx + D and the lower and upper limits of integration (a and b). It then:
- Determines the indefinite integral (antiderivative)
F(x)of the given polynomial. - Evaluates
F(x)at the upper limit (b) and the lower limit (a). - Subtracts
F(a)fromF(b)to find the definite integral value, according to the Fundamental Theorem of Calculus.
Example Calculation
Let's calculate the definite integral of f(x) = x² + 2x + 1 from x = 0 to x = 2.
- Function:
f(x) = 0x³ + 1x² + 2x + 1 - Coefficient A: 0
- Coefficient B: 1
- Coefficient C: 2
- Coefficient D: 1
- Lower Limit (a): 0
- Upper Limit (b): 2
First, find the indefinite integral F(x):
F(x) = (0/4)x⁴ + (1/3)x³ + (2/2)x² + 1x + C
F(x) = (1/3)x³ + x² + x + C
Now, apply the Fundamental Theorem of Calculus:
∫_0^2 (x² + 2x + 1) dx = F(2) - F(0)
F(2) = (1/3)(2)³ + (2)² + (2) = (1/3)(8) + 4 + 2 = 8/3 + 6 = 8/3 + 18/3 = 26/3
F(0) = (1/3)(0)³ + (0)² + (0) = 0
∫_0^2 (x² + 2x + 1) dx = 26/3 - 0 = 26/3 ≈ 8.666667
Using the calculator with these inputs should yield approximately 8.666667.