Calculate Riemann Sum

Riemann Sum Formula:

\[ \text{Riemann Sum} = \sum_{i=1}^{n} f(x_i) \times \Delta x \]

Function f(x):

Lower Bound (a):

Upper Bound (b):

Number of Intervals (n):

Method:

Riemann Sum:

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1. What is Riemann Sum?

The Riemann Sum is a method for approximating the total area under a curve (integral) by dividing the area into simple shapes (usually rectangles) and summing their areas. It's fundamental to integral calculus.

2. How Does the Calculator Work?

The calculator uses the Riemann Sum formula:

\[ \text{Riemann Sum} = \sum_{i=1}^{n} f(x_i) \times \Delta x \]

Where:

\( f(x_i) \) — Function value at sample point
\( \Delta x \) — Width of each subinterval (\( \frac{b-a}{n} \))
\( n \) — Number of subintervals
\( x_i \) — Sample point in subinterval (left, right, or midpoint)

Explanation: The area under the curve is approximated by summing the areas of rectangles with heights determined by the function value at specific points in each subinterval.

3. Importance of Riemann Sum

Details: Riemann Sums are the foundation of definite integrals in calculus. They are used to approximate areas, compute work done by variable forces, calculate probabilities in statistics, and more.

4. Using the Calculator

Tips: Enter a mathematical function of x (e.g., "x^2", "sin(x)", "2*x+3"), the interval bounds (a to b), number of subintervals, and select the method (left, right, or midpoint).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between left, right, and midpoint sums?
A: Left uses the function value at the left of each subinterval, right at the right, and midpoint at the center. Midpoint typically gives better approximations.

Q2: How many intervals should I use?
A: More intervals give better approximations but require more computation. Start with 10-100 for reasonable results.

Q3: Does this calculator support all mathematical functions?
A: This basic version supports simple PHP math expressions. For advanced functions, a proper parser would be needed.

Q4: How does Riemann Sum relate to definite integrals?
A: The definite integral is the limit of Riemann Sums as the number of subintervals approaches infinity.

Q5: Can I use this for functions of multiple variables?
A: No, this calculator is for single-variable functions only. Multiple variables require double or triple integrals.