1. M 112 Short Course in Calculus Chapter 5 – Accumulated Change: The Definite Integral Sections 5.2 – The Definite Integral V. J. Motto

2. Left- and Right-Hand Sums Figure 5.14: Left-hand sum: Area of rectangles Figure 5.15: Right-hand sum: Area of rectangles 10/7/2015

4. Problem 7 Using figure 5.21, draw rectangles representing each of the following Riemann sums for the function f on the interval 0 ≤ t ≤ 8. Calculate the value of each sum. (a) Left-hand sum with Δt = 4 (b) Right-hand sum with Δt = 4 (c) Left-hand sum with Δt = 2 (d) Right-hand sum with Δt = 2 Figure 5.21 10/7/2015

5. Solution to 7-a,b: (a) Left-hand sum = 32 · 4 + 24 · 4 = 224. (b) Right-hand sum = 24 · 4 + 0 · 4 = 96.

6. 10/7/2015 Solution to 7-c,d: (c) Left-hand sum = 32 · 2 + 30 · 2 + 24 · 2 + 14 · 2 = 200. (d) Right-hand sum = 30 · 2 + 24 · 2 + 14 · 2 + 0 · 2 = 136.

7. Problem 9 Use Figure 5.23 to estimate Figure 5.23 10/7/2015

8. Solution to 9: We know that f(x)dx = Area under f(x) between x = −10 and x = 15. The area under the curve consists of approximately 14 boxes, and each box has area (5)(5) = 25. Thus, the area under the curve is about (14)(25) = 350, so

9. 10/7/2015 Doing Integration with the Calculator: TI-83/84 Home Screen => Math =>9:fnInt( => fnInt(y1, x, start, end) TI-89 Home Screen => F3 => 2 ⌠( inegrate => ⌠(y19x0, start, end)