Calculate definite integrals ∫ab f(x) dx numerically using the trapezoidal rule. Enter mathematical expressions using either explicit or implicit multiplication, including x^2 + 2x, 3(x+1), and 2sin(x).
Enter a function and its integration limits.
Standard mathematical notation is supported, so you can write
x^2 + 2x instead of
x^2 + 2*x.
This calculator accepts common mathematical notation with implicit multiplication. You do not need to insert an asterisk between a coefficient and the variable x.
3x^2.
A definite integral calculates the accumulated value of a function over a specific interval. It is commonly written as:
Here, a is the lower limit, b is the upper limit, and f(x) is the function being integrated.
In geometric applications, the definite integral can represent the signed area between a curve and the x-axis. Portions of the function above the x-axis contribute positively, while portions below the x-axis contribute negatively.
This calculator uses the numerical trapezoidal rule. The interval is divided into 1,000 smaller sections and the function is evaluated at points throughout the interval.
For n subdivisions, the trapezoidal rule uses:
This calculator uses n = 1000 subdivisions. The result is therefore a numerical approximation rather than a symbolic antiderivative.
You can now enter the function without multiplication symbols:
For example, calculate:
Some definite integrals can be evaluated exactly by finding an antiderivative and applying the Fundamental Theorem of Calculus. Other functions are more conveniently evaluated numerically.
This calculator performs numerical integration using the trapezoidal rule. It does not attempt to produce a symbolic antiderivative.
If you need an antiderivative rather than a numerical value, use the Indefinite Integral Calculator .
Explore other integration tools on Summe.org:
Yes. This version of the calculator automatically interprets
2x as 2*x. Therefore,
x^2 + 2x and
x^2 + 2*x are both accepted.
Yes. The calculator recognizes implicit multiplication between
a number and parentheses, so 3(x+1) is interpreted
as 3*(x+1).
Yes. The calculator interprets x(x+1) as
x*(x+1).
Yes. Expressions such as 2sin(x) are recognized
as 2*sin(x).
No. This calculator uses numerical trapezoidal integration with 1,000 subdivisions, so the result is an approximation.
The calculation may fail if the function contains an undefined point, such as division by zero or a logarithm outside its real-valued domain.
A definite integral has lower and upper limits and produces a numerical value. An indefinite integral produces an antiderivative and normally includes a constant of integration.
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