The Fundamental Theorem of Calculus
Differentiation and integration are inverse operations. If you add up (integrate) a rate of change, you recover the total. This is the idea the rest of this topic builds on - everything below proves it.
In words: to find the definite integral of between and , find a function whose derivative is , then subtract its value at the lower limit from its value at the upper limit.
A stretched “S” for sum. We are summing infinitely many infinitely thin strips.
The lower and upper limits: the -values the area runs from and to.
The width of each strip. A strip has height and width , so area .
This is definite integration: it has limits and gives a number (an area). The boxes below build the theorem up from scratch - open them in order.
1. A circle: the integral of the circumference is the area
The area of a circle of radius is . Differentiate with respect to the radius:
That is exactly the circumference. Growing the radius by a tiny wraps a thin ring of length and width around the edge - an extra area of .
Summing every ring from the centre out to rebuilds the whole disc:
Integrating the circumference gives back the area. Area is the running total of its own rate of change.
2. A sphere: the integral of the surface area is the volume
The volume of a sphere is . Differentiate with respect to the radius:
That is the surface area. A tiny increase adds a thin shell of surface area and thickness , i.e. a volume .
Summing every shell from the centre out to rebuilds the solid ball:
Integrating the surface area gives back the volume - the same idea, one dimension up.
3. A straight line : area is a running total
Take the simplest sloped line, , and the area under it from to . It is a triangle, base 4 and height 4:
Now look at the area function that measures the area from up to any point . Its rate of change is
The rate at which area accumulates is exactly the height of the curve. And - the triangle area again.
4. A curve : strips close in on the true area
For a curve we cannot use a triangle. Instead split the area into rectangular strips. The more strips, the closer the total gets to the true area under between and (true value ).
As the strips get thinner the staircase hugs the curve and the total area tends to the exact value. Integration is the limit of this process as the strip width .
5. The proof: from first principles
Let be the area under up to . Add a thin strip of width between and .
- The extra area is roughly a rectangle, height , width :
- Divide by - and this is the gradient of :
- Let the strip become infinitely thin, . The approximation becomes exact:
This is the first half of the theorem: the area function differentiates back to the curve. Integration undoes differentiation.
6. Why the limits subtract: the cancels
Any function with derivative has the form for some constant . The area between and is the area up to minus the area up to :
The appears in both brackets and cancels. That is why a definite integral needs no constant of integration - and why we write
You have now built the Fundamental Theorem of Calculus. The Definite Integration tab puts it to work.
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.
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.
Question
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.
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 curves equal to each other and solve - the solutions are the -coordinates where they cross.
- Top minus bottom. Decide which curve is higher in that interval and integrate the difference:
- Evaluate using the Fundamental Theorem of Calculus. Subtracting first keeps the answer positive even below the axis.