Numerical Integration Calculator
Approximate definite integrals numerically using Simpson's Rule, the Trapezoidal Rule, Left and Right Riemann Sums, or the Midpoint Rule.
What Is Numerical Integration?
Numerical integration is a group of methods used to approximate the value of a definite integral when an exact antiderivative is difficult, impossible, or unnecessary to calculate.
For a definite integral \[ \int_a^b f(x)\,dx, \] the interval from \(a\) to \(b\) is divided into smaller subintervals. The function is then sampled at selected points and the resulting values are combined to estimate the area under the curve.
This numerical integration calculator provides several common methods so you can compare different approximations for the same integral.
Numerical Integration Methods
Trapezoidal Integration
The Trapezoidal Rule approximates the area under a curve by replacing each section of the curve with a straight line. Each section therefore forms an approximate trapezoid.
It is useful for general numerical integration and is straightforward to apply to many functions.
Simpson's Rule
Simpson's Rule uses quadratic approximations rather than straight-line approximations. It generally uses pairs of subintervals and therefore requires an even number of subintervals.
If an odd value of \(n\) is entered, this calculator automatically increases it by one so Simpson's Rule can be applied.
Left Riemann Sum
The Left Riemann Sum evaluates the function at the left endpoint of each subinterval and uses those values to approximate the integral.
The approximation can be useful for understanding how Riemann sums relate to definite integrals.
Right Riemann Sum
The Right Riemann Sum evaluates the function at the right endpoint of each subinterval. Like the Left Riemann Sum, its accuracy depends on the function and the number of subintervals.
Midpoint Rule
The Midpoint Rule evaluates the function at the midpoint of every subinterval and multiplies the sum of those values by the subinterval width.
How to Use the Numerical Integration Calculator
- Enter the function \(f(x)\), such as
x^2 + 3*x. - Enter the lower integration limit \(a\).
- Enter the upper integration limit \(b\).
- Choose the number of subintervals \(n\).
- Select a numerical integration method.
- Click Calculate Numerical Integral to see the approximation.
Increasing the number of subintervals generally produces a finer approximation, although the behavior depends on the function and numerical method being used.
Supported Function Notation
The calculator accepts common mathematical notation, including:
x^2for \(x^2\)x^3 + 2xfor \(x^3+2x\)2(x+1)sin(x)
cos(x)tan(x)sqrt(x)ln(x)
For multiplication, you can use either * or common implicit
multiplication such as 2x and 2(x+1).
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Numerical Integration FAQ
What is numerical integration?
Numerical integration estimates the value of a definite integral using sampled values of a function instead of finding an exact antiderivative.
What is the Trapezoidal Rule?
The Trapezoidal Rule approximates the area under a curve by dividing the integration interval into sections and approximating each section with a trapezoid.
Why does Simpson's Rule require an even number of subintervals?
Simpson's Rule works with pairs of subintervals. For this reason, the total number of subintervals must be even. If an odd number is entered, this calculator increases it by one.
Does increasing \(n\) make numerical integration more accurate?
Increasing the number of subintervals generally gives the numerical method more points with which to approximate the function. However, accuracy depends on the function, interval, numerical method, and behavior of the function across the interval.
What is the difference between a Riemann sum and numerical integration?
Riemann sums are a fundamental way of approximating an integral by adding areas of rectangles. Numerical integration includes Riemann sums as well as other methods such as the Trapezoidal Rule, Simpson's Rule, and the Midpoint Rule.