Integral Calculator
Free Integral Calculator for calculus. Enter values to get step-by-step solutions with formulas and graphs. See charts, tables, and visual results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Integral Calculator
Calculator
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Formula: Simpson's Rule: โซf(x)dx โ (h/3)[f(xโ) + 4f(xโ) + 2f(xโ) + 4f(xโ) + ... + f(xโ)]
Worked example โ 0.33333333
Formula
Simpson's Rule: โซf(x)dx โ (h/3)[f(xโ) + 4f(xโ) + 2f(xโ) + 4f(xโ) + ... + f(xโ)]
Simpson's 1/3 rule uses quadratic interpolation for numerical integration. It requires an even number of subintervals. Error is O(hโด), making it much more accurate than the trapezoidal rule for smooth functions.
Worked Examples
Example 1: โซxยฒ from 0 to 1
Problem:f(x) = xยฒ, bounds [0, 1]
Solution:โซxยฒdx = xยณ/3 |โยน = 1/3 = 0.33333333
Result:0.33333333
Example 2: โซsin(x) from 0 to ฯ
Problem:f(x) = Math.sin(x), bounds [0, 3.14159265]
Solution:โซsin(x)dx = [-cos(x)] from 0 to ฯ = (-cos(ฯ)) - (-cos(0)) = 1 - (-1) = 2
Result:โ 2.00000000
Frequently Asked Questions
What is the difference between a definite and indefinite integral?
A definite integral has upper and lower bounds and produces a numerical result representing the net area under a curve. An indefinite integral produces an antiderivative function plus a constant C, representing the general family of functions whose derivative is the integrand. Integral Calculator evaluates definite integrals numerically.
Why might a numerical integral give an incorrect answer?
Singularities (where the function is undefined or infinite, like 1/x at x=0) cause numerical methods to fail or give wildly inaccurate results. Functions with very steep gradients or rapid oscillations may also require many more intervals for accuracy. Always verify results against known antiderivatives when possible.
What real-world quantities are computed using definite integrals?
Definite integrals compute the total distance traveled from a velocity function, the total work done by a variable force, the total charge delivered by a varying current over time, the probability of a continuous random variable falling in a range (area under a probability density function), and the total revenue from a continuously varying price/demand function.
Can Integral Calculator evaluate improper integrals (with infinite bounds)?
No โ Integral Calculator requires finite numeric upper and lower bounds, since Simpson's rule divides a finite interval into subintervals. Improper integrals with infinite bounds, or with a singularity at an endpoint, require specialized techniques (often approximated using a substitution to transform the interval, or a suitably large finite bound as an approximation) beyond straightforward Simpson's rule evaluation.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator ยท Editorial policy
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