1. Section 5.2
Definite Integrals

2. Definite Integral and Area
All continuous functions are integrable.
If y = f(x) is nonnegative and integrable over a closed interval [a, b], then the area under the curve y = f(x) from a to b is the integral of f from a to b.
A = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
(It doesn’t matter which letter you use to denote the variable of integration).

3. The Integral Sign
All together, this statement is read “the integral from a to b of f of x dx.”

4. Negative Area If f(x) < 0, then Area = - 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
For any integrable function, 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = area above the x-axis – area below the x-axis.
Use approximation methods to find (𝑥 −3) 2 − 3 𝑑𝑥

5. Integrals on the calculator
We have already discussed finding an integral using Riemann sums and using geometric methods.
To calculate a definite integral on the calculator, use the fnInt command found in the “math” menu.
Use this command to calculate integrals that you estimated using Riemann sums.

6. Constant Function
If f(x) = c, where c is a constant on the interval [a,b], then 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 = ?
= 𝑎 𝑏 𝑐 𝑑𝑥 = c (b – a)