Free maths tool · Level: Ages 17 to 18
Integral and Riemann sum
An integral is an area before it is anything else. Push the rectangle count up and watch the staircase become a smooth curve.
A definite integral from a to b is the area between the graph and the x axis. You approximate it by chopping the region into rectangles and adding their areas, which is a Riemann sum. More rectangles means less error, and the integral is where those sums are heading.
Why the sign matters
Below the x axis the rectangles are drawn in a second colour and they subtract rather than add. An integral is a signed area, not an ordinary one.
Take sin(x) from 0 to 2π and you land on exactly zero. The hump above the axis and the hump below cancel out. Taking "integral equals area" too literally loses marks right here.
Left, right, midpoint or trapezium
For a rising function the left sum always sits below and the right sum always above. The true value is trapped between them, which is a handy check on its own.
Set n to 20 and switch between the four methods. Midpoint and trapezium land far closer than left or right at the same chop. More rectangles helps, measuring smarter helps faster.
Where the dx comes from
Each rectangle has area f(x) times its width. In the sum that width is written Δx. Let the number of pieces run to infinity and Δx becomes infinitely small, which we write as dx.
That is the whole story behind ∫ f(x) dx: a stretched S for "sum", with height times width after it.
Integrals still not clicking?
Book a 1-on-1 lesson with a verified maths tutor, whenever it suits you.
Frequently asked questions
- What does an integral actually mean?
- The signed area between the graph and the x axis, between two bounds. Above the axis it adds, below the axis it subtracts.
- What is a Riemann sum?
- An approximation of that area using a row of rectangles. You measure the height at the left, right or middle of each strip, multiply by the width, and add them all up.
- Why are more rectangles more accurate?
- Each rectangle has a flat top and the curve does not, so a sliver is always left over. Narrower rectangles leave less room for that error, and the gap shown on the right shrinks with it.
- How does this relate to antiderivatives?
- The fundamental theorem says that limit of sums can also be worked out as F(b) - F(a), where F is an antiderivative of f. This tool shows the area side of that story, the rules are the other side.