Numerical Integration
Approximate definite integrals using the Trapezoidal Rule
Use JavaScript syntax: x * x (for x²), Math.sin(x), Math.exp(x), etc.
- Enter a function f(x) using JavaScript syntax
- Specify the lower and upper bounds of integration
- Choose the number of intervals (more intervals = higher precision)
- The calculator approximates the definite integral using the Trapezoidal Rule
- The Trapezoidal Rule approximates the area under the curve by dividing it into trapezoids
Common Functions:
x * x for x²
Math.sin(x) for sin(x)
Math.cos(x) for cos(x)
Math.exp(x) for e^x
Math.sqrt(x) for √x
Formula: The Trapezoidal Rule approximates ∫f(x)dx by summing the areas of trapezoids formed between function values.
What is numerical integration?
Numerical integration is a technique to approximate the value of a definite integral when an analytical solution is difficult or impossible to find. It divides the area under a curve into smaller shapes (like trapezoids) whose areas can be easily calculated and summed.
What is the Trapezoidal Rule?
The Trapezoidal Rule approximates the area under a curve by dividing it into trapezoids. For each subinterval, it connects the function values at the endpoints with a straight line, then calculates the trapezoid's area. It's simple to implement and works well for smooth functions.
How does increasing intervals improve accuracy?
More intervals create smaller trapezoids that better approximate the curve. The error in the Trapezoidal Rule decreases proportionally to 1/n^2, where n is the number of intervals. Doubling the intervals roughly quadruples the accuracy, though this comes at increased computational cost.
When should I use numerical integration?
Use numerical integration when you can't find an antiderivative analytically, when dealing with experimental data points, or when the function is too complex for symbolic integration. It's essential in physics, engineering, and scientific computing for solving real-world problems.