Integral Sum Calculator
'/%3E%3Ccircle cx='48' cy='39' r='19' fill='%23f2c9a5'/%3E%3Cpath d='M27 35c3-16 39-17 42 2-8-8-27-9-42-2z' fill='%23374151'/%3E%3Cpath d='M21 86c5-24 49-28 56 0z' fill='%232563eb'/%3E%3Ccircle cx='41' cy='41' r='2' fill='%23111827'/%3E%3Ccircle cx='55' cy='41' r='2' fill='%23111827'/%3E%3Cpath d='M42 52c4 3 9 3 13 0' stroke='%2392400e' stroke-width='3' fill='none' stroke-linecap='round'/%3E%3C/svg%3E)
Reviewed by Calculator Editorial Team
The Integral Sum Calculator helps you compute the sum of integrals for functions over specified intervals. This tool is essential for physics, engineering, and mathematical analysis where understanding the accumulation of quantities is crucial.
What is Integral Sum?
An integral sum, also known as a Riemann sum, is a method used to approximate the definite integral of a function. It works by dividing the area under the curve into small rectangles, summing their areas, and taking the limit as the number of rectangles approaches infinity.
This concept is fundamental in calculus and has applications in physics, engineering, and economics. The integral sum provides a way to calculate the total accumulation of quantities that vary continuously over an interval.
How to Calculate Integral Sum
Calculating the integral sum involves several steps:
- Define the function you want to integrate.
- Specify the interval [a, b] over which you want to calculate the integral.
- Choose the number of subintervals (n) to divide the interval.
- Calculate the width of each subinterval (Δx = (b - a)/n).
- Evaluate the function at the left endpoint of each subinterval.
- Multiply each function value by Δx to get the area of each rectangle.
- Sum all the rectangle areas to get the approximate integral value.
For more accurate results, increase the number of subintervals. The integral sum converges to the exact integral value as n approaches infinity.
Formula
The integral sum formula is:
Σ f(xi) Δx, where i ranges from 0 to n-1
With Δx = (b - a)/n and xi = a + iΔx
This formula represents the sum of the areas of rectangles under the curve of the function f(x) from a to b, divided into n equal parts.
Examples
Example 1: Linear Function
Calculate the integral sum for f(x) = x from 0 to 2 with n = 4.
Δx = (2-0)/4 = 0.5
x0 = 0, x1 = 0.5, x2 = 1, x3 = 1.5
Sum = f(0)*0.5 + f(0.5)*0.5 + f(1)*0.5 + f(1.5)*0.5 = 0 + 0.25 + 0.5 + 0.75 = 1.5
Example 2: Quadratic Function
Calculate the integral sum for f(x) = x² from 1 to 3 with n = 5.
Δx = (3-1)/5 = 0.4
x0 = 1, x1 = 1.4, x2 = 1.8, x3 = 2.2, x4 = 2.6
Sum = f(1)*0.4 + f(1.4)*0.4 + f(1.8)*0.4 + f(2.2)*0.4 + f(2.6)*0.4 ≈ 1 + 2.744 + 5.776 + 9.68 + 17.156 ≈ 36.356
FAQ
What is the difference between integral sum and definite integral?
The integral sum is an approximation of the definite integral. As the number of subintervals increases, the integral sum approaches the exact value of the definite integral.
How does increasing the number of subintervals affect the result?
Increasing the number of subintervals makes the approximation more accurate. The integral sum converges to the exact integral value as n approaches infinity.
Can I use this calculator for any function?
Yes, you can use this calculator for any continuous function. Simply input the function, interval, and number of subintervals to get the integral sum.