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Chapter 5 | Integration
Figure 5.15 The graph of y = sin x is divided into six regions: Δ x = π /2 6 = π 12 .
⎡ ⎣ π
⎤ ⎦ ,
⎡ ⎣ π 6
⎤ ⎦ ,
⎡ ⎣ π
⎤ ⎦ ,
⎡ ⎣ π
⎤ ⎦ , and
⎡ ⎣ 0, π
⎤ ⎦ ,
⎡ ⎣ 5 π
⎤ ⎦ . Note that f ( x ) = sin x is
π 6
, π 4
π 3
5 π 12
π 2
The intervals are
12
12 ,
4 ,
3 ,
12 ,
increasing on the interval ⎡
⎤ ⎦ , so a left-endpoint approximation gives us the lower sum. A left-endpoint
⎣ 0, π 2
approximation is the Riemann sum ∑ i =0 5
⎛ ⎝ π
⎞ ⎠ . We have
sin x i
12
⎛ ⎝ π 6
⎞ ⎠
⎛ ⎝ π 3
⎞ ⎠
A ≈ sin(0) ⎛
⎞ ⎠ +sin
⎛ ⎝ π
⎞ ⎠
⎛ ⎝ π
⎞ ⎠ +sin
⎛ ⎝ π
⎞ ⎠ +sin
⎛ ⎝ π 4
⎞ ⎠
⎛ ⎝ π
⎞ ⎠ +sin
⎛ ⎝ π
⎞ ⎠ +sin
⎛ ⎝ 5 π 12
⎞ ⎠
⎛ ⎝ π
⎞ ⎠
⎝ π
12
12
12
12
12
12
12
=0.863.
Using the function f ( x ) = sin x over the interval ⎡ ⎣ 0, π 2 ⎤
5.6
⎦ , find an upper sum; let n =6.
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