Calculus / Unit 8: The Fundamental Theorem of Calculus
8.1 FTC Part 2: Evaluating Definite Integrals
Notes
Fundamental Theorem of Calculus, Part 2: if $F' = f$ on $[a, b]$ (and $f$ is continuous), then $\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)$.
- Find an antiderivative (no $+ C$ needed), then subtract: top value minus bottom value.
- Notation: $\left[F(x)\right]_a^b = F(b) - F(a)$.
- Example: $\displaystyle \int_1^3 2x\,dx = \left[x^2\right]_1^3 = 9 - 1 = 8$.
Practice
Find $\displaystyle \int_1^3 2x\,dx$.
An antiderivative is $x^2$. Then $9 - 1 = 8$.
Find $\displaystyle \int_1^4 \sqrt{x}\,dx$.
An antiderivative is $\displaystyle \frac{2}{3}x^{3/2}$. Then $\displaystyle \frac{2}{3}(8 - 1) = \frac{14}{3}$.
In FTC Part 2, why is $+ C$ not needed?
$(F(b) + C) - (F(a) + C) = F(b) - F(a)$.
Ready for more? Take the Unit 8 test.