Enter the width of the segment and the function values at the left, middle, and right endpoints into the calculator to determine the approximation of the definite integral using Simpson's 1/3 Rule.

Simpson's 1/3 Rule Formula

The following formula is used to calculate the approximation of a definite integral using Simpson's 1/3 Rule. 

I = (h/3) * (y0 + 4y1 + y2)

Variables:

  • I is the approximation of the definite integral
  • h is the width of the segment y0, y1, and y2 are the function values at the left, middle, and right endpoints of the segment, respectively

To calculate the approximation of a definite integral using Simpson's 1/3 Rule, first determine the width of the segment. Then, find the function values at the left, middle, and right endpoints of the segment. Multiply the width of the segment by 1/3, then multiply this result by the sum of the function value at the left endpoint, four times the function value at the middle endpoint, and the function value at the right endpoint. The result is the approximation of the definite integral.

What is a Simpson's 1/3 Rule?

Simpson's 1/3 Rule is a numerical integration technique used to approximate definite integrals. It works by dividing the area under a curve into a series of parabolic arcs, then summing the areas of these arcs to estimate the total area. The rule is named "1/3" because each segment's area is calculated as (h/3) * (y0 + 4y1 + y2), where h is the width of the segment and y0, y1, and y2 are the function values at the left, middle, and right endpoints of the segment, respectively. This method is more accurate than the Trapezoidal Rule for functions that are not linear.

How to Calculate Simpson's 1/3 Rule?

The following steps outline how to calculate the Simpson's 1/3 Rule.

  1. First, determine the width of the segment (h).
  2. Next, determine the function values at the left endpoint (y0), middle endpoint (y1), and right endpoint (y2).
  3. Next, gather the formula from above = I = (h/3) * (y0 + 4y1 + y2).
  4. Finally, calculate the approximation of the definite integral (I).
  5. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

width of the segment (h) = 0.5

function value at the left endpoint (y0) = 2

function value at the middle endpoint (y1) = 3

function value at the right endpoint (y2) = 4