1. Definition of Integral

2. Riemann Sum
Sum given by the formula

3. Integral
If f is defined on the interval [ a, b] and the limit exists, then this limit is called the definite integral of f from a to b

4. f(x) is called the integrand a is the lower limit b is the upper limit

5. Example
Express the Riemann sum below as an integral on the interval [ 0, π ]

6. Example
Evaluate the integral below using the limit method 0 3 𝑥 2 𝑑𝑥

7. Homework
Evaluate using the limit method 1) 0 3 5𝑑𝑥 2) 0 2 2𝑥𝑑𝑥 3) 𝑥 2 𝑑𝑥

8. Areas Using Geometry
Using the region corresponding to each definite integral, evaluate 1) 2) 3)

9. Properties of Integrals

10. Example
1) 2)

11. Example
If it is known the and find