Area Under Curve Calculator

Formula first

Overview

This formula represents the Second Fundamental Theorem of Calculus, which provides a computational method for evaluating definite integrals. It defines the net area under a curve as the difference between the values of the function's antiderivative evaluated at the upper and lower limits of integration.

Symbols

Variables

A = Area, F(b) = Upper Limit Val, F(a) = Lower Limit Val

Apply it well

When To Use

When to use: Use this formula when calculating the accumulated change of a continuous function over a specific interval [a, b]. It is applicable whenever an antiderivative F(x) can be identified for the integrand f(x) such that F'(x) = f(x).

Why it matters: This relationship is the foundation of integral calculus, allowing scientists to solve complex problems in physics, engineering, and economics. It turns the geometric problem of finding areas into a straightforward algebraic calculation of evaluation.

Avoid these traps

Common Mistakes

• Order of subtraction (F(a)-F(b)).
• Forgetting to integrate first.

One free problem

Practice Problem

A particle moves along a path where the antiderivative of its velocity function represents its position. If the position at the end of the journey (Fb) is 50 meters and the position at the start (Fa) is 15 meters, calculate the total displacement (A) representing the area under the velocity curve.

Solve for: $A$

Hint: Subtract the initial antiderivative value from the final antiderivative value.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Calculus: Early Transcendentals by James Stewart
2. Wikipedia: Fundamental theorem of calculus
3. Thomas' Calculus
4. Halliday, Resnick, and Walker, Fundamentals of Physics
5. Stewart, J. (2016). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
6. Thomas, G. B., Weir, M. D., & Hass, J. (2018). Thomas' Calculus (14th ed.). Pearson.
7. AQA A-Level Mathematics — Pure (Integration)