A definite integral represents the limit of Riemann sums as the rectangle width approaches . The core setup uses and sample points in each subinterval. Left, right, and midpoint sums use different sample points, so their estimates can be overestimates or underestimates depending on the function.
The Fundamental Theorem of Calculus connects this limit process to antiderivatives through when .
Key Facts
- For equal subintervals on , the width is .
- A general Riemann sum is , where is a sample point in the th subinterval.
- The left Riemann sum is .
- The right Riemann sum is .
- The midpoint Riemann sum is .
- The definite integral is when the limit exists.
- If , then the Fundamental Theorem of Calculus gives .
- The net area counts area above the -axis as positive and area below the -axis as negative.
Vocabulary
- Riemann sum
- A sum that approximates the signed area under a curve using rectangles.
- Definite integral
- The limit of Riemann sums on an interval, written , representing net accumulation.
- Subinterval
- One smaller interval formed when is divided into pieces, each with width for equal partitions.
- Sample point
- The chosen input inside a subinterval where the function height is measured.
- Net area
- The signed area found by adding positive regions above the -axis and negative regions below the -axis.
- Antiderivative
- A function whose derivative is , so .
Common Mistakes to Avoid
- Using the wrong width is incorrect because the rectangle width must be for equal subintervals.
- Mixing left and right endpoints is incorrect because uses while uses .
- Forgetting negative area is incorrect because measures net area, so regions where subtract from the total.
- Treating as always positive is incorrect because definite integrals can be negative, zero, or positive depending on signed accumulation.
- Applying with the wrong antiderivative is incorrect because the Fundamental Theorem requires .
Practice Questions
- 1 Compute for divided into equal subintervals, then list the left endpoints.
- 2 Use a right Riemann sum with to approximate .
- 3 Evaluate using an antiderivative.
- 4 For an increasing positive function on , explain whether or is larger and why.
Understanding Riemann Sums & The Definite Integral
A Riemann sum is a model of accumulation. Each rectangle combines a small input change with an output value. Its height comes from the function, while its width comes from a small part of the interval.
Adding the rectangle values gives an estimate of the total change. This idea works even when the graph does not describe physical area. If a graph shows velocity in metres per second against time in seconds, each rectangle represents a small displacement in metres.
Adding them estimates how far the object moved. If a graph shows water flow in litres per minute against time, the accumulated value is a volume of water.
The shape of the graph helps predict estimation error before any arithmetic. For an increasing function, left endpoint rectangles tend to sit below the curve, while right endpoint rectangles tend to rise above it. For a decreasing function, that pattern reverses.
Midpoint rectangles often give a closer estimate because the midpoint can better represent the average height across a short interval. Concavity matters too. A curve that bends upward may make midpoint estimates too small, while a curve that bends downward may make them too large.
These patterns are useful checks, not replacements for calculation. A graph can change direction or curvature, so students should inspect each section rather than applying one rule to the whole interval without thought.
Signed accumulation is one of the most important ideas. A function value below the horizontal axis contributes a negative amount because it represents change in the opposite direction. For velocity, positive and negative contributions produce displacement, not total distance travelled.
A person can walk forward for part of a trip and backward later. Their displacement may be zero even though they travelled a positive distance.
To find total distance from velocity, split the interval where the velocity changes sign and treat each piece as positive before adding. This distinction appears in many settings, including profit and loss, electric charge flow, and changes in population.
The connection to antiderivatives is powerful because it replaces a long approximation process with endpoint evaluation when an antiderivative is known. It works because an antiderivative records accumulated change in a precise way. Still, numerical sums remain important.
Many real data sets come from measurements at separate times, not from a neat formula. A fitness tracker estimates distance from recorded speeds. Environmental scientists estimate rainfall volume from rainfall rates sampled during a storm.
In these cases, the choice of time interval affects accuracy. Shorter intervals usually reduce error, but measurements can still contain noise.
When learning this topic, track units at every step. Function units multiplied by horizontal-axis units must match the units of the accumulated result. Keep the interval endpoints in the correct order, since reversing them reverses the sign.
Make a table when there are many rectangles. List each subinterval, the chosen sample point, its function value, and its contribution. On graph problems, draw the rectangles or mark the sample points.
On calculator problems, do not trust a decimal answer alone. Decide whether it represents net change, total positive area, or an estimate with a known possible error.