What integration is
Integration is a way of adding up lots of tiny pieces to find a total — most often the total area underneath a curve. The neat part: integration is the exact opposite of differentiation, so to integrate you just ask "what would I have to differentiate to get this?"
To find the area under a curve between and :
- Un-differentiate the curve: find a function that differentiates back to . (So .)
- Put in the end value, then the start value, and subtract: the area is .
It looks scary, but every symbol just labels a part of the recipe above. Here is what each one means:
A stretched “S” for sum. It says: add up all the thin strips of area.
The -values the area runs from (, the bottom number) and to (, the top number).
One thin strip has height and a tiny width , so its area is height × width .
Because it has a start and an end, this is called definite integration — it gives an actual number (an area). The examples below show why the recipe works, starting from the simplest picture. Open whichever ones you like.
1. The simplest case: area under the straight line
What this shows: for a shape we already know, integration gives the same answer as ordinary geometry — so we can trust it.
The area under from to is just a triangle. Using base × height:
Now do it the integration way. We need a function that differentiates to ; that is (check: differentiating gives ). Then put in the end, put in the start, and subtract:
Same answer, 8. The two methods agree.
2. A curve : why we add up strips
What this shows: a curve is not a triangle, so we cover the area with thin rectangle strips and add them up. Use the + button to add more strips and watch the total close in on the true area under from to (which is ).
The thinner the strips, the better the staircase matches the curve. Integration is what you get when the strips become infinitely thin — the estimate becomes the exact area.
3. Why "un-differentiating" works:
What this shows: the area you have built up, , differentiates straight back to the curve . That is the link between area and differentiation.
Let stand for the area under the curve up to the point . Now nudge along by a tiny width to , adding one more strip (shaded green).
- That extra strip is almost a rectangle: height , width , so its area is about :
- Divide both sides by . The left side is now exactly the formula for the gradient of :
- Shrink the strip until . The "about equal" becomes "exactly equal":
So differentiating the area gives the curve. Going the other way — integrating the curve — gives the area. They undo each other.
4. Why we subtract start from end: the cancels
What this shows: there are many functions that differentiate to (they differ by a constant ), but when you subtract, the disappears — so it never affects an area.
Any "un-differentiated" function can be written . The area from to is "area up to " minus "area up to ":
The is in both square brackets, so it cancels. This is why a definite integral never needs the , and why we write:
That is the whole recipe. The Definite Integration tab puts it to work on real examples.
5. The same idea in a circle: integrate the circumference to get the area
What this shows: the "add up thin strips" idea is everywhere, not just under graphs. Here the strips are thin rings.
A circle of radius has area . Differentiating with respect to the radius:
which is exactly the circumference. Growing the radius a little wraps a thin ring (length , width ) round the edge. Add up every ring from the centre out to and you rebuild the whole disc:
Integrating the circumference gives back the area.
6. And in a sphere: integrate the surface area to get the volume
What this shows: the very same idea, one dimension up — the strips are now thin shells.
A sphere of radius has volume . Differentiating with respect to the radius:
which is the surface area. Each tiny increase in radius adds a thin shell of volume . Add up every shell out to and you rebuild the solid ball:
Integrating the surface area gives back the volume.
Families of curves and finding the exact one
Integrating gives a whole family of curves, all the same shape but shifted up or down by the constant . One extra point pins down which member of the family you want.
Type a point the curve passes through:
Method
A gradient function tells you the gradient everywhere but not the height - exactly like knowing the gradient of a straight line without knowing its intercept in . One known point fixes .
- Integrate the gradient function to get the general solution, remembering :
- Substitute the known point to form an equation in - just like using to lock a line onto a point.
- Solve for and write out the one particular curve.
Questions
Estimating area with rectangles
Some functions cannot be integrated by hand - such as . We can still estimate the area underneath by slicing it into rectangles. Using the left height of each strip and the right height brackets the true area between an under- and an over-estimate.
Method — how to calculate it
Each rectangle has the same width (the gap between ordinates). Its height is the -value at one edge.
- Find the strip width: .
- Find the ordinates by substituting into .
- Add the heights and multiply by the width. Left rectangles use the left edge of each strip, right rectangles use the right edge:
For an increasing curve the left rectangles sit under the curve (an under-estimate) and the right rectangles poke above it (an over-estimate). The true area lies between them.
The trapezium rule
Rectangles waste the corner triangles. Joining the tops of the ordinates with straight lines makes trapeziums that fit the curve far better - and generalising the working gives the trapezium rule.
Where the rule comes from (general proof)
The area of one trapezium is . The parallel sides are two neighbouring ordinates and the width is the strip width . Adding four strips out individually:
Every interior ordinate () is shared by two trapeziums, so it appears twice; the two ends () appear once. Writing out each strip separately is silly - we generalise to any number of strips:
"Ends once, middles twice, all times ." That is the trapezium rule.
Questions
Definite integration
A definite integral evaluates to a number. Integrate, write the result in square brackets with the limits, substitute the upper then the lower limit, and subtract. Watch the sign: area below the -axis comes out negative.
Questions
Try each one yourself first, then open the working.
The area between two curves
The area trapped between two curves is the integral of the top curve minus the bottom curve, taken between the -values where they cross. Subtracting first means it works even when the region sits below the axis.
Method
- Find the limits. Set the two equations equal to each other and solve - the answers are the -coordinates where the curves cross.
- Top minus bottom. Decide which curve is higher between those points and integrate the difference:
- Evaluate with the Fundamental Theorem of Calculus (bracket, substitute, subtract). Subtracting top minus bottom first keeps the answer positive, even if the region dips below the axis.