Integration as Area Under a Curve: From Rectangles to the Definite Integral
Build the definite integral visually from rectangle approximations and understand width, height, limits and signed area.
Quick answer
A definite integral adds infinitely many extremely thin contributions f(x)dx between two limits. Rectangle sums approximate that accumulation; as the rectangles become narrower, the sum approaches the exact signed area represented by the integral.
Key relationships
Approximate the area with rectangles
Divide the interval from a to b into smaller pieces. Over each piece, choose a sample x-value and use f(x) as the rectangle height. Multiply that height by the interval width Δx, then add every rectangle's contribution.
Left-endpoint, right-endpoint and midpoint choices can give different estimates when only a few rectangles are used. As the maximum width approaches zero under suitable conditions, these sums converge to the same definite integral.
Read every part of the integral symbol
The limits a and b identify the interval. The function f(x) supplies the local height or rate. The dx names the variable of integration and represents the limiting idea of very small widths.
The elongated integral sign grew from the idea of a sum. It is a compact instruction to accumulate local contributions continuously across an interval.
Separate signed area from geometric area
Where f(x) is above the x-axis, contributions are positive. Where it is below, contributions are negative. A definite integral can therefore be zero even when the graph encloses visible regions, because positive and negative parts cancel.
To calculate ordinary geometric area, split the interval at x-axis crossings and add the absolute sizes of the pieces. The Fundamental Theorem then evaluates each definite integral using an antiderivative.
What to remember
- Each rectangle contributes height × width.
- Narrower partitions usually improve the approximation.
- A definite integral measures signed accumulation, not always ordinary geometric area.
- The Fundamental Theorem connects accumulation to antiderivatives.
Frequently asked questions
Is an integral always an area?
No. It represents accumulation and can describe displacement, mass, probability and many other quantities. Area is the most common visual model.
Why do thinner rectangles help?
A narrow interval allows the rectangle height to follow changes in the function more closely, reducing over- and under-estimation.
Why can area under the x-axis be negative?
The definite integral is signed: values of f(x) below the axis contribute negative amounts. Geometric area uses absolute values instead.
Watch rectangles turn into a definite integral
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