1. What is the Displacement Formula in Calculus?

The displacement formula \( s_{\text{calc}} = \int_{a}^{b} f(x) dx \) calculates the net displacement of an object by integrating its position function f(x) over a time interval [a, b]. This represents the area under the velocity curve or the net change in position.

2. How Does the Calculator Work?

The calculator uses numerical integration to approximate the definite integral:

Where:

• \( f(x) \) — Position function (meters)
• \( a \) — Lower limit of integration (start time)
• \( b \) — Upper limit of integration (end time)
• \( dx \) — Infinitesimal change in x

Explanation: The calculator approximates the integral using numerical methods (Riemann sums) to compute the net displacement between time a and time b.

3. Importance of Displacement Calculation

Details: Calculating displacement is fundamental in physics and engineering for determining an object's net position change, analyzing motion, and solving problems involving velocity and acceleration.

4. Using the Calculator

Tips: Enter the lower and upper limits (a and b), and the position function f(x) using standard mathematical notation. Use 'x' as the variable and standard operators (+, -, *, /, ^ for exponentiation).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between displacement and distance?
A: Displacement is a vector quantity (net position change with direction), while distance is a scalar quantity (total path length traveled).

Q2: Can I use any mathematical function?
A: The calculator supports basic operations. For complex functions, consider using specialized mathematical software.

Q3: How accurate is the numerical integration?
A: Accuracy depends on the number of intervals used. More intervals provide better accuracy but require more computation.

Q4: What if my function has discontinuities?
A: The calculator may not handle discontinuities well. For piecewise functions, break the integral into separate continuous intervals.

Q5: Can I use this for velocity functions?
A: Yes, the integral of velocity gives displacement, and the integral of acceleration gives velocity.