Calculator Online

Home AI Chat AI Math Solver

NEW

Tools ▾ Category ▾

Pricing Sign In

Calculator Online

Your Result is copied!

Simpson's Rule Calculator

Choose between Simpson’s ⅓ or ⅜ rule, and select your input: function or data table. Then enter the required values and click on “Calculate” to determine the area under the curve.

Using:

Simpson 1/3

Simpson 3/8

Method:

Enter a Function:

Upper Limit:

Lower Limit:

Number of rectangles n:

Related

Our Simpson’s Rule calculator helps you approximate definite integrals step by step. Choose between Simpson's ⅓ or ⅜ Rule to estimate the area under a curve. The calculator also works with tables of data points, making it versatile for both functions and discrete datasets. Just select your preferred method, input the values, and let the calculator do the rest!

Note: Accuracy depends on the number of points and their spacing. Generally, more points improve the result.

What is Simpson’s Rule?

Simpson’s Rule is a numerical method for approximating definite integrals. Instead of rectangles, it uses parabolas to approximate the area under a curve. Named after Thomas Simpson, it often provides higher accuracy than simpler methods like the trapezoidal rule.

The main idea: given three points on a curve, a unique quadratic (parabola) can be drawn through them. Simpson’s Rule then sums these parabolic areas for the integral estimate.

Versions of Simpson’s Rule

There are two commonly used forms:

Simpson’s 1/3 Rule – uses sets of three points (two subintervals)
Simpson’s 3/8 Rule – uses sets of four points (three subintervals) for slightly improved accuracy

Simpson’s 1/3 Rule Formula:

\(\int_a^b f(x) dx \approx \frac{h}{3} \left[ f(a) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(b) \right]\)

or in terms of data points \(y_i\):

\(\int_a^b f(x) dx \approx \frac{\Delta x}{3} \left[ y_0 + 4(y_1 + y_3 + \dots) + 2(y_2 + y_4 + \dots) + y_n \right]\)

\(h = \frac{b - a}{n}\) — width of each subinterval
n — number of subintervals (must be even)
\(f(a)\) and \(f(b)\) — function values at endpoints

Note: Subintervals must be equally spaced and even in number.

Simpson’s 3/8 Rule Formula:

\(\int_a^b f(x) dx \approx \frac{3h}{8} \left[ f(a) + 3 \sum f(x_{\text{other points}}) + 2 \sum f(x_{\text{every third}}) + f(b) \right]\)

The coefficients for n+1 points follow the repeating pattern:

\(1, 3, 3, 2, 3, 3, 2, ..., 3, 3, 1\)

Note: The number of subintervals must be a multiple of 3.

Steps to Apply Simpson’s Rule

Identify the interval endpoints \(a\) and \(b\)
Decide the number of subintervals \(n\)
Compute subinterval width: \(h = (b-a)/n\)
Divide the interval into n equal parts
Apply Simpson’s 1/3 or 3/8 formula to approximate the integral

Example:

Approximate the integral of \(y = 3^x\) from x = 0 to x = 1 using Simpson’s 1/3 Rule with n = 2.

Step 1: \(\Delta x = (1-0)/2 = 0.5\)

Step 2: Subinterval points: 0, 0.5, 1

Step 3: Evaluate function: \(f(0) = 1\), \(f(0.5) \approx 1.732\), \(f(1) = 3\)

Step 4: Apply Simpson’s 1/3 Rule:
\(\int_0^1 3^x dx \approx \frac{0.5}{3}[1 + 4(1.732) + 3] \approx 2.004\)

Exact value: \(\int_0^1 3^x dx = \frac{3-1}{\ln 3} \approx 1.8205\)

Limitations

Oscillatory functions: Rapid oscillations reduce accuracy
Discontinuous functions: May yield incorrect estimates
Endpoint issues: Vertical asymptotes or abrupt changes at interval edges can cause errors