What it shows
A definite integral ∫ₐᵇ f(x) dx is the signed area between the graph of f and the x-axis from a to b: area above the axis counts plus, area below counts minus. Once you see it that way, the properties in your syllabus stop being rules to memorise and become things you can see. Pick a property, pick a function, move the limits, and the page draws both sides and works both out, so you can watch them agree.
- Swap the limits: going from b back to a flips the sign.
- Split at c: the area from a to b is the area from a to c plus the area from c to b, even when c lies outside [a, b].
- f(a + b − x): this is the graph flipped left to right inside [a, b], so its area is the same. It is the trick behind many clever integrals, such as ∫₀^(π/2) sin x/(sin x + cos x) dx = π/4.
- From 0 to 2k: the second half can be flipped onto the first as f(2k − x).
- Even and odd functions: on [−k, k], an even function’s halves are equal, so the integral is twice the right half; an odd function’s halves cancel, so it is 0.
- Area function: A(x) = ∫ₐˣ f(t) dt grows at the rate f(x). This is the first fundamental theorem of calculus.
It follows integrals in Class 12 maths (definite integrals and their properties, the fundamental theorem). Before this page, riemann sums shows where the area comes from; for the antiderivatives you need to work integrals by hand, derivative steps runs the rules backwards in your head, and e and ln covers the eˣ and ln x you meet in them.
How to use it
- Pick a property from the first list.
- Pick f(x): x², x³ − x, sin x, cos x, |x|, eˣ, x sin x, 2x + 1 or sin² x, or choose My own… and type one, such as
x^2 cos(x). - Move the sliders that the property needs: a and b (the limits), c (the split point), k (the half-width) or x (the end of the area function).
- Read the numbers under the picture: both sides of the property, worked out to four decimals, and a line saying whether they agree.
Worked example
Pick Even and odd functions and x³ − x with k = 1.5. The right half ∫₀^1.5 (x³ − x) dx = 1.5⁴/4 − 1.5²/2 = 1.2656 − 1.125 = 0.1406, and the left half is −0.1406. The whole integral is 0, for any k, because x³ − x is odd. Switch to x² with k = 1.5: the halves are both 1.125 and the whole is 2.25 = 2 × 1.125.
Pick Split at a point c with x² from a = −0.5 to b = 2 and c = 0.75. The pieces are 0.1823 and 2.526, which add to 2.7083 = (8 + 0.125)/3, the whole integral.
FAQ
How does the page work the integrals out? By Simpson’s rule with 2000 strips, which is accurate to many decimals for these smooth functions. The checks compare it with exact answers from a computer algebra system.
Why is |x| even but not smooth? |x| has a corner at 0, but |−x| = |x|, so it is even and the shortcut works.
Is this page part of NCERT? No. fibub is independent. It follows the topics of the Class 12 maths syllabus and uses none of the textbook’s text or exercises.
Where the numbers come from
- Standard properties of definite integrals and the first fundamental theorem of calculus. Written for fibub; no textbook text, figures or exercises are used.
- Checked against sympy 1.14 (integrate): five integrals each of nine functions, the parity of each, and ∫₀^(π/2) sin x/(sin x + cos x) dx = π/4; integrals by composite Simpson's rule with 2000 strips.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.