Area Under the Curve
Calculating the area bounded by a function and the x-axis
The area under a curve is one of the most fundamental applications of definite integrals. It allows us to calculate the exact area bounded by a function, the x-axis, and vertical lines.
Definition
The area under a curve y = f(x) from x = a to x = b is given by the definite integral:
Geometric Interpretation
The definite integral can be interpreted as the signed area between the curve y = f(x) and the x-axis from x = a to x = b:
- If f(x) ≥ 0 on [a, b], the integral gives the area between the curve and the x-axis.
- If f(x) ≤ 0 on [a, b], the integral gives the negative of the area between the curve and the x-axis.
- If f(x) changes sign on [a, b], the integral gives the algebraic sum of the areas.
Examples
Example 1: Area under a parabola
Example 2: Area with a negative function
Applications
The concept of area under a curve has numerous applications in various fields:
Physics
- Work done by a variable force
- Distance traveled from a velocity-time graph
- Electric charge from a current-time graph
Economics
- Consumer and producer surplus
- Income distribution (Lorenz curves and Gini coefficient)
- Total cost from marginal cost
Probability
- Probability distributions
- Expected values
- Cumulative distribution functions
Common Mistakes
- Forgetting to check if the function is negative in some regions
- Not identifying where the function crosses the x-axis
- Confusing the area under the curve with the definite integral when the function takes negative values
- Using incorrect limits of integration