Definite integral visualizer
A definite integral ∫ₐᵇ f(x)dx isn't magic — it's the limit of a sum of thin rectangles as they get thinner and thinner. So build the rectangles and watch. Drag N up and the Riemann sum creeps toward the exact area, computed independently from the antiderivative.
The ideas you're seeing
What a definite integral means
∫ₐᵇ f(x)dx is defined as the limit of a Riemann sum — thin rectangles — as the number of rectangles goes to infinity and their width goes to zero. "Area under the curve" is just a picture of that limit.
The Fundamental Theorem of Calculus
Instead of computing that limit directly, find an antiderivative F(x) where F′(x)=f(x), then just compute F(b) − F(a). This shortcut turns an infinite sum into simple substitution — it's why calculus is useful.
More rectangles, less error
As N grows, the rectangles fit the curve's shape more and more tightly and the approximation error shrinks toward zero. Slide N up and watch the sum converge to the exact value.
Why the midpoint rule is used here
Using f at each rectangle's midpoint — rather than its left or right edge — balances out the over- and under-estimates from the curve's bend, converging to the true area faster than left/right sums for the same N.
Part of the Integrals chapter — read the notes, grab the formula sheet and take the quiz. One of Priodemy for School, free with every EduSuite school.
From rectangles to the Fundamental Theorem
The definite integral is a limit, not a formula
Before it is anything else, ∫ab f(x) dx is defined as a limit of sums. Slice the interval from a to b into n strips, approximate each strip by a rectangle, add the areas, then let n grow without bound. As the rectangles narrow, the staircase hugs the curve more closely and the sum converges on the exact area. Increase n here and watch the error shrink — the definition is the animation.
This matters because the definition explains why an integral can represent so many different things. If f is a velocity, the strips are small distances and the integral is total displacement. If f is a rate of flow, the integral is total volume. Anything built by accumulating a quantity over an interval is an integral, and area is just the most convenient picture.
Why the Fundamental Theorem is remarkable
Computing that limit directly is painful. The Fundamental Theorem of Calculus supplies a shortcut that looks almost too good: if F is any antiderivative of f, then ∫ab f(x) dx = F(b) − F(a). An infinite summation collapses into two substitutions and a subtraction.
The reason it works is that differentiation and integration are inverse operations. The rate at which accumulated area grows, as you extend the right-hand edge, is exactly the height of the curve at that edge. That is the link between the two halves of calculus, and it is why the same chapter teaches both.
Signed area, and what the sign means
Where the curve dips below the x-axis, the rectangles have negative height and contribute negatively. The definite integral therefore returns signed area, which can be zero even for a curve that is nowhere flat — integrate sin x from 0 to 2π and the positive and negative halves cancel exactly.
This is why "find the area enclosed" and "evaluate the integral" are different questions. For genuine area you must split the interval at every point where the curve crosses the axis, integrate each piece, and add the absolute values.
Mistakes that cost marks
Adding +C to a definite integral. The constant cancels in F(b) − F(a), so it belongs only to indefinite integrals. Writing it in a definite answer signals that the two have been confused.
Subtracting in the wrong order. It is upper limit minus lower limit. Reversing them flips the sign, and swapping the limits of an integral is exactly what negates it.
Forgetting to change limits after substitution. When you substitute u = g(x), the limits are values of x and must be converted to values of u. Carrying the original limits through is one of the most frequent errors in this chapter, and the resulting answer is usually plausible enough to go unnoticed.