Using the Definite Integral Calculator
The definite integral calculator is a specialized tool designed to solve calculus problems involving the accumulation of quantities and the area under a curve. Unlike indefinite integrals, which result in a general function (plus a constant C), a definite integral results in a specific numerical value representing the net signed area between the function and the x-axis within a specified interval [a, b].
This calculator focuses on polynomial functions, which are the most common foundation for learning calculus principles like the Power Rule and the Fundamental Theorem of Calculus.
- Upper Limit (b)
- The right-hand boundary of the interval of integration.
- Lower Limit (a)
- The left-hand boundary of the interval of integration.
- Coefficients (A, B, C, D)
- The numerical multipliers for each term in the polynomial function Ax³ + Bx² + Cx + D.
How It Works
The calculator utilizes the Fundamental Theorem of Calculus (Part 2), which states that if f(x) is continuous on [a, b] and F(x) is an antiderivative of f(x), then:
∫ab f(x) dx = F(b) – F(a)
For a polynomial function, we apply the Power Rule for Integration to each term:
- ∫ xⁿ dx = (xⁿ⁺¹) / (n + 1)
- The constant of integration C is omitted because it cancels out when subtracting F(b) – F(a).
- The "net area" means that area above the x-axis is positive and area below is negative.
Calculation Example
Example: Calculate the definite integral of f(x) = 3x² + 2x from x = 1 to x = 3.
Step-by-step solution:
- Identify coefficients: A=0, B=3, C=2, D=0. Limits: a=1, b=3.
- Find antiderivative F(x): ∫(3x² + 2x) dx = x³ + x².
- Evaluate at Upper Limit: F(3) = (3)³ + (3)² = 27 + 9 = 36.
- Evaluate at Lower Limit: F(1) = (1)³ + (1)² = 1 + 1 = 2.
- Subtract: 36 – 2 = 34.
- Result = 34
Common Questions
What happens if the lower limit is greater than the upper limit?
If you swap the limits of integration, the result will have the opposite sign. Mathematically, ∫ab f(x) dx = -∫ba f(x) dx.
Does this definite integral calculator work for trig functions?
This specific version is optimized for polynomials. For trigonometric functions like sin(x) or cos(x), different integration rules (like ∫sin(x) dx = -cos(x)) would need to be programmed into the solver logic.
Is the result always positive?
No. If the function lies mostly below the x-axis within the given interval, the definite integral will be a negative number. This represents a "negative area" relative to the axis.