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
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Figure 5.15: Right-hand sum:
Area of rectangles
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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
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Solution to 7-a,b:
(a) Left-hand sum = 32 · 4 + 24 · 4 = 224.
(b) Right-hand sum = 24 · 4 + 0 · 4 = 96.
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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.
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Problem 9
Use Figure 5.23 to estimate
Figure 5.23
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Solution to 9:
We know that

15
10
f ( x)dx
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

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15
10
f ( x)dx  350
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)
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