1. Section 4.1 Areas and Distances Math 1231: Single-Variable Calculus

2. Areas: Example Example Use rectangles to estimate the area under the parabola y = x 2 from 0 to 1.

3. Areas: Example Right sum Left sum

4. Areas: Example

5. Areas: General case

6. Area: General Case Right Sum: R n = f(x 1 )Δx + f(x 2 )Δx + … + f(x n )Δx A = lim n  ∞ R n = lim n  ∞ [ f(x 1 )Δx + f(x 2 )Δx + … + f(x n )Δx ] A = lim n  ∞ L n = lim n  ∞ [ f(x 0 )Δx + f(x 1 )Δx + … + f(x n-1 )Δx ]

7. Sigma notation

8. Distance Problem Example Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30-second time interval. We take speedometer readings every five seconds and record them in the following table:

9. Distance Problem: If we wanted a more accurate estimate, we could have taken velocity readings every two seconds, or even every second.