Use polynomial format: 3x^2 + 2x - 5, x^3 - 1, etc.
∫ab f(x) dx = F(b) − F(a)
F(x) = Antiderivative of f(x)
a = Lower limit of integration
b = Upper limit of integration
∫01 x^2 dx = 1/3
∫02 (3x^2 + 2x) dx = 12
∫14 x dx = 7.5
A definite integral is a fundamental concept in calculus that represents the signed area between a function and the x-axis over a specific interval [a, b]. Unlike indefinite integrals, which give a family of functions, definite integrals produce a single numerical value. This value can represent physical quantities like area, volume, displacement, work, or accumulated quantities.
The definite integral is evaluated using the Fundamental Theorem of Calculus, which connects differentiation and integration. It states that if F(x) is an antiderivative of f(x), then the definite integral from a to b equals F(b) − F(a). This powerful theorem allows us to compute areas and accumulated quantities efficiently.
To calculate a definite integral, follow these steps: First, find the indefinite integral (antiderivative) of the function using integration rules like the power rule, sum rule, and constant multiple rule. Second, evaluate the antiderivative at the upper limit (b) and lower limit (a). Finally, subtract the lower evaluation from the upper evaluation to get the definite integral value.
Step 1: Find the Antiderivative
Apply integration rules to find F(x), the antiderivative of f(x). For polynomials, use the power rule: increase the exponent by 1 and divide by the new exponent.
Step 2: Evaluate at Limits
Substitute the upper limit (b) into F(x) to get F(b), and the lower limit (a) to get F(a).
Step 3: Subtract
Calculate the final answer by computing F(b) − F(a). This gives the definite integral value.
Definite integral calculations follow standard calculus rules. Results depend on correct function input, variable selection, and limit values. This calculator supports polynomial functions. For more complex functions involving trigonometric, exponential, or logarithmic terms, results may require verification using specialized mathematical software.