Plot a function \(f(x)\), visualize the signed area between the curve and the x-axis, and approximate the definite integral \(\int_a^b f(x)\,dx\) directly in your browser.
Enter mathematical expressions naturally. For example, x^2+2x, 2(x+1), sin(x), sqrt(x), or ln(x).
Graphical integration connects the visual graph of a function with the numerical value of its definite integral. For a function \(f(x)\) on an interval from \(a\) to \(b\), the definite integral is written as \[ \int_a^b f(x)\,dx. \]
Geometrically, the integral represents signed area between the graph and the x-axis. Area above the x-axis contributes positively, while area below the x-axis contributes negatively. This distinction is important when interpreting the numerical result of an integral.
This graphical integration calculator plots the function and estimates the definite integral using the composite trapezoidal rule.
The interval \([a,b]\) is divided into a number of smaller subintervals. Instead of calculating the exact area under the curve, each small section is approximated by a trapezoid.
If the points are evenly spaced, the step size is \[ h=\frac{b-a}{n-1}. \]
Increasing the number of points generally makes the graphical approximation more detailed and can improve the numerical approximation when the function is sufficiently smooth.
The calculator accepts several forms of ordinary mathematical notation, so you do not always need to insert multiplication symbols explicitly.
x^2 + 2x
(x+1)(x-1)
2(x+1)
2sin(x) or xcos(x)
exp(x)
ln(x)
sqrt(x)
Enter:
x^2
For the interval \([0,2]\), the integral is \[ \int_0^2 x^2\,dx = \frac{8}{3} \approx 2.6667. \]
Enter:
2x+1
The calculator recognizes 2x as multiplication,
so you do not have to write 2*x.
Enter:
sin(x)
When a function crosses the x-axis, the portions below the axis contribute negatively to the signed integral.
A common source of confusion is the difference between a definite integral and the total geometric area between a curve and the x-axis.
is a signed quantity. If \(f(x)\) is negative on part of the interval, that portion subtracts from the result.
By contrast, total geometric area is related to \[ \int_a^b |f(x)|\,dx. \]
The result is a numerical approximation rather than a symbolic antiderivative. Its accuracy depends on the function, the integration interval, and the number of graph points.
Smooth functions usually work well with the trapezoidal rule, particularly when the interval is divided into sufficiently small pieces. Functions with sharp curvature, discontinuities, singularities, or rapid oscillations may require considerably more points or a different integration method.
For an integral where you want a numerical answer without the graphical display, you can also use the Numerical Integration Calculator.
For symbolic antiderivatives, use the Indefinite Integral Calculator.
Integration is closely connected to differentiation through the Fundamental Theorem of Calculus. Studying the graph of a function alongside its derivative and integral can make this relationship easier to understand.
It plots \(f(x)\) over the selected interval and estimates the definite integral \(\int_a^b f(x)\,dx\) using the composite trapezoidal rule.
Yes. The calculator supports common implicit multiplication,
including x^2+2x, 2(x+1),
x(x-1), and expressions such as
2sin(x).
The trapezoidal rule approximates the area under a curve by replacing small portions of the curve with straight-line trapezoids and summing their signed areas.
Yes. The calculator computes the signed definite integral. Values of \(f(x)\) below the x-axis contribute negatively.
Increasing the number of points gives a finer numerical approximation and can improve accuracy for smooth functions. However, increasing the number of points does not eliminate problems caused by discontinuities or singularities.
Some functions are undefined for particular real x-values. For example, \(\ln(x)\) is not defined for \(x\leq0\) in the real numbers, and \(\sqrt{x}\) requires \(x\geq0\). The entire selected interval must therefore contain valid real-valued function evaluations.
No. This page performs numerical integration using sampled graph points and the trapezoidal rule. For symbolic antiderivatives, use the Indefinite Integral Calculator.
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