Definite Integral Calculator

Calculate the area under a curve between two specific points (limits of integration).

Integration Settings

Function f(x)

Lower Limit (a)

Upper Limit (b)

Note on Syntax

Use x as the variable. Standard math operators like +, -, *, / are supported. Use ^ for powers (e.g., x^2). Functions like sin(x), cos(x), exp(x) are also available.

Integration Result

2.6667

Area/Integral Value

Lower Bound0

Upper Bound2

Graph of f(x) and Integrated Area

The shaded region represents the area being calculated (integral from 0 to 2).

About this calculator

What is a Definite Integral?

In calculus, a definite integral represents the signed area of the region in the xy-plane bounded by the graph of f(x), the x-axis, and the vertical lines x=a and x=b. It calculates the cumulative sum of infinitesimal rectangles across an interval.

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Pro Tips

The lower limit 'a' can be greater than the upper limit 'b', which flips the sign of the result.
If the function falls below the x-axis, that area is subtracted from the total.
Definite integrals are widely used to find volumes, work, and center of mass.

Fun Facts

"Archimedes used a method of exhaustion to find areas, which was a precursor to modern integration."
"The symbol ∫ is actually an elongated 'S', standing for 'summa' (Latin for sum)."
"Definite integrals are the mathematical engine behind calculating the total energy consumed over time."

Formula: ∫[a, b] f(x) dx = F(b) - F(a)