Medium

Area Under a Curve

Simple Explanation

When f(x) ≥ 0 on [a,b], the definite integral ∫ₐᵇf(x)dx gives exactly the area of the region trapped between the curve y=f(x) and the x-axis, from x=a to x=b.

Why Do We Need It?

This is the most direct geometric meaning of a definite integral, and the starting point for every other application in this chapter — area between curves and volumes of revolution both build directly on it.

See It

The area under y = x², between x = 1 and x = 2

Formula

Area Under a Curve

Area = ∫ₐᵇ f(x) dx (for f(x) ≥ 0 on [a,b])

The definite integral of a non-negative function over [a,b] equals the exact area trapped between the curve and the x-axis over that interval.

f(x): — the function whose graph forms the top boundary of the region
a, b: — the left and right x-boundaries of the region

When to use it: Whenever the exact area between a curve and the x-axis, over a specific interval, is needed.

Worked Example

Find the area under a curve

Find the area under y = x² between x = 1 and x = 2.

Why Does This Work?

A definite integral is defined as the limit of a sum of infinitely many, infinitely thin rectangles of height f(x) and width dx spanning [a,b] — when f(x)≥0, that sum of rectangle areas is exactly the total area under the curve, and the Fundamental Theorem of Calculus provides a shortcut to compute that limit exactly.

Real-Life Example

Total water collected from a variable-rate inflow pipe

A pipe fills a tank at a rate (in liters per minute) that changes over time, described by a function r(t).

The total volume collected between two times is exactly the area under the rate function's graph over that time interval — found by evaluating the definite integral of r(t).

Practice

Find the area under y = 3x² between x = 0 and x = 2.

Medium

Common mistake

Using this formula directly when f(x) is negative somewhere on [a,b] — the plain integral would then subtract that region's area instead of adding it, so a genuinely negative section needs to be split off and its absolute value taken separately.

Quick Review

Area = ∫ₐᵇ f(x)dx, valid directly when f(x)≥0 on [a,b].
Find an antiderivative, then apply the Fundamental Theorem of Calculus.
The foundation for area-between-curves and volume-of-revolution problems.