What is a Definite Integral Calculator?
A definite integral calculator computes the integral of a function over a specified interval [a, b]. It is used to find the area under a curve, accumulated quantities in physics, and total change in economics. This tool simplifies complex manual calculations, making it essential for students, engineers, and researchers needing quick and accurate results.
Definite Integral Formula
The definite integral of a function f(x) from a to b is calculated using the Fundamental Theorem of Calculus: ∫ab f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
How to Use This Calculator
1. Enter the function (e.g., x^2, sin(x)).
2. Input lower and upper limits.
3. Click "Calculate". The result will display the integral value.
4. Use "Clear" to reset inputs. FAQs appear after calculation.
FAQs
1. What is the purpose of a definite integral calculator?
A definite integral calculator computes the area under a curve between two points, aiding in physics, engineering, and economics. It automates complex integration, saving time and reducing errors in manual calculations.
2. How accurate is this calculator?
This calculator uses numerical methods (Riemann sums) for approximation. Accuracy depends on the function complexity and interval division. Results are precise for most educational purposes.
3. Can it handle trigonometric functions?
Yes. Enter functions like sin(x), cos(x), or tan(x). Use JavaScript syntax (e.g., Math.sin(x)). Ensure correct spelling and parentheses.
4. What if my function has discontinuities?
The calculator may give errors if the function is discontinuous or undefined in [a, b]. Check the function’s continuity before calculating.
5. Why does my result show "Error"?
Invalid functions (e.g., 2x instead of 2*x), syntax errors, or non-numeric limits cause errors. Review your input and use explicit multiplication (*).
6. Can I use variables other than x?
No. This calculator only processes functions of x. Replace other variables with x for correct results.
7. How is the result derived mathematically?
The calculator approximates using Riemann sums: dividing [a, b] into small intervals, calculating areas of rectangles, and summing them. More intervals increase accuracy.
8. Is this tool suitable for educational use?
Yes. It helps students verify answers and understand integration concepts. However, learning manual integration remains crucial for exams.
9. Can I compute improper integrals?
No. This calculator handles finite intervals only. For improper integrals (infinite limits), use specialized tools.
10. Are multivariable functions supported?
No. This calculator only supports single-variable functions (x). For multivariable calculus, use partial integral tools.
Examples of Definite Integrals
| Function | Lower Limit | Upper Limit | Result |
|---|---|---|---|
| x^2 | 0 | 2 | 8/3 ≈ 2.6667 |
| sin(x) | 0 | π | 2 |
| 3*x + 5 | 1 | 4 | 28.5 |
Derivation Process
The definite integral is derived from the Fundamental Theorem of Calculus. First, find the antiderivative F(x) of f(x). Evaluate F(x) at the upper limit b and subtract its value at the lower limit a. This theorem links differentiation and integration, enabling precise area calculations.