The Fundamental Theorem of Calculus
The fundamental theorem of calculus links derivatives and integrals: \(\displaystyle\int_a^b f(x)\,dx=F(b)-F(a)\), where \(F'=f\).
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Integral calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
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.