Numerical Integral Calculator

What is a definite integral?

A definite integral measures the area between a curve and the x-axis, from a starting value a to an ending value b. If a derivative is about the slope at a point, an integral is about the accumulated total across a range. It answers questions like the total distance covered when you know the speed at every moment.

This calculator finds that area numerically using Simpson's rule, which fits small parabolic arcs to the curve. Parabolas hug a curve more closely than straight lines, so the estimate is accurate even with relatively few segments.

How to use this calculator

1. Enter the function you want to integrate, such as \(x^2+1\) or \(\sin(x)\).
2. Type the lower limit \(a\) and upper limit \(b\) of the definite integral.
3. Choose or confirm the number of subintervals, which must be even for Simpson's rule.
4. Click calculate to get the approximate integral value and, if needed, compare it with the displayed steps or formula.

The formula explained

The definite integral \(\int_a^b f(x)\,dx\) computes the signed area between the graph of \(f(x)\) and the x-axis from \(a\) to \(b\). Simpson's rule estimates that area by fitting parabolas over pairs of subintervals, which often gives a better approximation than simple rectangles.

• f(x) = the function being integrated
• a = the lower limit of integration
• b = the upper limit of integration
• n = the number of equal subintervals, which must be even for Simpson's rule
• h = the width of each subinterval, \(h=(b-a)/n\)

Step by step method

1. Split the interval \([a,b]\) into an even number \(n\) of equal parts, so each part has width \(h=(b-a)/n\).
2. Evaluate the function at the endpoints and at each interior point \(x_1, x_2, \dots, x_{n-1}\).
3. Apply Simpson's rule: \(\int_a^b f(x)\,dx \approx \frac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+\cdots+4f(x_{n-1})+f(x_n)]\).

Worked example

Problem. Approximate \(\int_0^2 (x^2+1)\,dx\) using Simpson's rule with \(n=2\).

1. Compute the step size, \(h=(2-0)/2=1\).
2. Find the function values: \(f(0)=1\), \(f(1)=2\), and \(f(2)=5\).
3. Use Simpson's rule: \(\frac{1}{3}[1+4(2)+5]=\frac{14}{3}\approx 4.667\).

Tips and common mistakes

• Simpson's rule requires an even number of subintervals, so if your calculator asks for \(n\), choose 2, 4, 6, and so on.
• If the function changes quickly or has a sharp turn, using more subintervals usually improves the approximation.

Frequently asked questions