M 112 Short Course in Calculus

Chapter 5 – Accumulated Change: The Definite Integral Sections 5.2 – The Definite Integral V. J. Motto

Left- and Right-Hand Sums

Figure 5.14: Left-hand sum: Area of rectangles 4/30/2020 Figure 5.15: Right-hand sum: Area of rectangles

4/30/2020

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 Δ

t

= 4 (b) Right-hand sum with Δ

t

= 4 (c) Left-hand sum with Δ

t

= 2 (d) Right-hand sum with Δ

t

= 2 Figure 5.21

4/30/2020

Solution to 7-a,b: 4/30/2020 (a) Left-hand sum = 32 · 4 + 24 · 4 = 224.

(b) Right-hand sum = 24 · 4 + 0 · 4 = 96.

Solution to 7-c,d: 4/30/2020 (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.

Problem 9

Use Figure 5.23 to estimate 4/30/2020 Figure 5.23

Solution to 9: We know that 15   10 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 15   10  350 4/30/2020

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) 4/30/2020