Integral Calculator
Integral Calculator computes the definite integral of a function over an interval, e.g., \\( \int_0^1 x^2 \, dx \\). Ideal for calculus students, teachers, and professionals.
About the Integral Calculator
The Integral Calculator computes the definite integral \\( \int_a^b f(x) \, dx \\) of a function over an interval using numerical integration (Simpson’s Rule). It supports polynomials, trigonometric functions, exponentials, logarithms, and constants like \\( \pi \\) or \\( e \\), making it ideal for calculus, physics, and engineering.
- Enter a function and interval (e.g., x^2; [0, 1] or sin(x); [0, pi]) using the keypad or keyboard.
- Press Calculate to view the integral value, steps, and a graph of the function.
- Use Clear to reset the input to empty or Backspace (⌫) to delete the last character.
- Copy results or share via Facebook, WhatsApp, X.com, or embed the calculator.
How to Use the Integral Calculator
Follow these steps:
- Enter Function: Input a function (e.g., x^2, sin(x)) followed by a semicolon and interval (e.g., [0, 1]).
- Use Keypad: Click buttons for digits (0-9, .), variable (x), operators (+, -, *, /, ^), functions (sin, cos, ln, e^), constants (π), and interval symbols ([, ], ,).
- Correct Input: Use Backspace (⌫) to delete the last character or Clear to empty the input.
- Calculate: Press the “Calculate” button to display the integral, steps, and graph.
- Copy/Share: Copy results or share via social media or embed code.
Notes:
- Ensure the function is continuous over the interval.
- Use * for multiplication (e.g., 2*x) and ^ for powers (e.g., x^2).
- Interval format: [a, b] with a semicolon (e.g., x^2; [0, 1]).
- Keyboard support: Numbers, operators, Enter for calculation, Backspace to delete.
Integral Calculation Explained
The definite integral \\( \int_a^b f(x) \, dx \\) represents the area under the curve of \\( f(x) \\) from \\( x = a \\) to \\( x = b \\). This calculator uses Simpson’s Rule for numerical approximation:
Simpson’s Rule:
\\( \int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right] \\)
where \\( h = \frac{b – a}{n} \\), and \\( n \\) is even (default 1000).
Applications:
- Calculus: Compute areas under curves.
- Physics: Calculate work or displacement.
- Engineering: Analyze probability distributions.
Example Calculation
Calculate \\( \int_0^1 x^2 \, dx \\):
- Enter Input: Clear the input and type “x^2; [0, 1]”.
- Calculate: Press “Calculate”.
Steps: Integrate \\( x^2 \\) from 0 to 1 using Simpson’s Rule (1000 subintervals).
Result: \\( \approx 0.333333 \\)
Graph: Shows \\( f(x) = x^2 \\) with the area under the curve shaded. - Result: Integral: 0.333333.
Available Features
- Functions: Supports polynomials (x^2), trigonometric (sin, cos), exponentials (e^x), logarithms (ln), and constants (π, e).
- Input: Single field for function and interval, fully clearable.
- Results: Integral value, calculation steps, and graphical visualization.
- History: Stores up to 5 recent calculations.
- Sharing: Share via Facebook, WhatsApp, or X.com.
- Embedding: Copy embed code for external sites.
- Accessibility: ARIA attributes and keyboard support.
Calculator Functionality
- Error Handling: Validates function and interval, displays errors for invalid inputs.
- Result Formatting: Rounds to six decimal places.
- Responsive Design: Adapts to all screen sizes.
- Graphing: Visualizes the function and area under the curve.