The Fundamental Theorem of Calculus
Learn the Fundamental Theorem of Calculus for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). This result connects the two halves of calculus: the area-accumulating integral and the derivative are reverse processes, so areas can be found from primitives.
You will learn to evaluate a definite integral by finding any primitive and subtracting its value at the lower limit from its value at the upper limit, use square-bracket notation, and see why the constant cancels — the shortcut that replaces long sums of strips.
Theory
The Fundamental Theorem of Calculus links area and antiderivatives. This Year 12 Mathematics Advanced topic (MAV-12-05) evaluates \(\int_a^b f\,dx=F(b)-F(a)\).
If \(F'(x)=f(x)\), then \(\displaystyle\int_a^b f(x)\,dx=[F(x)]_a^b=F(b)-F(a)\).
Find any primitive \(F\) (no \(+C\) needed, it cancels), substitute the top limit, then subtract the bottom limit.
Method
- Find a primitive \(F\).
- Substitute the upper limit \(F(b)\).
- Subtract the lower limit \(F(a)\).
| \(\ \) | \(=\) | \(\big[\tfrac{x^{3}}{3}\big]_0^2\) |
| \(=\) | \(\tfrac{8}{3}\) |
| \(\ \) | \(=\) | \(\big[x^{2}\big]_1^2\) |
| \(=\) | \(4-1=3\) |
| \(\ \) | \(=\) | \(\big[x^{3}\big]_0^1=1\) |
| \(\ \) | \(=\) | \(\big[x^{2}+x\big]_1^3\) |
| \(=\) | \(12-2=10\) |
Common pitfalls
Frequently asked questions
What is the Fundamental Theorem of Calculus?
It says that to evaluate a definite integral you find an antiderivative F and compute F of b minus F of a.
Do you need plus C for a definite integral?
No. The constant cancels when you subtract F of a from F of b, so it is not needed.
How do you evaluate a definite integral?
Find a primitive, substitute the upper limit, then subtract the value at the lower limit.
What does the square bracket notation mean?
The notation with a bracket and limits means substitute the top limit into the primitive and subtract the value at the bottom limit.