1. Section 4.3 Riemann Sums and Definite Integrals

2. To this point, anytime that we have used the integral symbol we have used it without any upper or lower boundaries. In other words, it has served as simply a short hand symbol asking you to antidifferentiate a function. We have also been talking about areas simultaneously. Those problems do have upper and lower bound limits. There is a reason that we have been pursuing these two seemingly different goals at the same time.

3. Section 4.3 Riemann Sums and Definite Integrals The reason why is that they are the same goal, not different ones. Using words to describe a problem:  Find the area of the region bounded by the curve,the x -axis, and the lines x = 2 and x = 6. Using summation to describe the problem Using integral notation to describe the problem

4. Section 4.3 Riemann Sums and Definite Integrals Below is a visual representation of the region bounded by the curve and the vertical boundaries.

5. Section 4.3 Riemann Sums and Definite Integrals Draw the region defined by the integral below:

6. Section 4.3 Riemann Sums and Definite Integrals Did your graph look like the one below?

7. Section 4.3 Riemann Sums and Definite Integrals Now try to read a summation notation and turn it into a picture AND into a definite integral. Here’s the sigma notation:

8. Section 4.3 Riemann Sums and Definite Integrals Does your picture look like this?

9. Section 4.3 Riemann Sums and Definite Integrals Does your integral look like this?