The Definite Integral & Riemann Sums
Learn the definite integral and Riemann sums for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). Cutting the region under a curve into thin strips and adding their areas gives an estimate of the area, and the estimate improves as the strips get narrower.
You will learn to build rectangle sums, see the definite integral as the limit of those sums, explain why a region below the \(x\)-axis gives a negative value, and find some integrals exactly from simple shapes — the idea behind total distance from a velocity graph.
Theory
The definite integral is the signed area between a curve and the \(x\)-axis. This Year 12 Mathematics Advanced topic (MAV-12-05) introduces Riemann sums, which approximate that area with rectangles.
\(\displaystyle\int_a^b f(x)\,dx\) is the signed area from \(x=a\) to \(x=b\). Splitting \([a,b]\) into strips of width \(\Delta x\), each rectangle has area \(f(x_i)\,\Delta x\), and the Riemann sum \(\sum f(x_i)\,\Delta x\) estimates the total.
As the strips get thinner the sum approaches the exact integral. Area below the axis is negative.
Method
- Divide \([a,b]\) into strips of equal width.
- Add the rectangle areas \(f(x_i)\,\Delta x\).
- Refine with more strips for a better estimate (or integrate for the exact value).
| \(\ \) | \(=\) | \(1(1)+1(4)\) |
| \(=\) | \(5\) |
| \(\ \) | \(=\) | \(\tfrac12(2)(2)=2\) |
| \(\ \) | \(=\) | \(2\times2=4\) |
| \(\ \) | \(=\) | \(\tfrac12(3)(3)=4.5\) |
Common pitfalls
Frequently asked questions
What is a definite integral?
It is the signed area between a curve and the x-axis between two limits a and b, written as the integral from a to b of f of x dx.
What is a Riemann sum?
It is an approximation of the area under a curve by adding up the areas of thin rectangles, height times width, across the interval.
What does signed area mean?
Area above the x-axis counts as positive and area below counts as negative, so the integral gives the net, or signed, area.
How do you make a Riemann sum more accurate?
Use more, thinner rectangles. As the width approaches zero the sum approaches the exact definite integral.