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Chapter 4: Q. 55 (page 386)

For each function f and interval [a, b] in Exercises 53鈥�55, approximate the average value of f from x = a to x = b, using a sample size of at least eight.
$f (x) = \sin x, [0, 2 蟺]$

Short Answer

Expert verified

The average value of f from x = a to x = b, using a sample size of at least eight is $0 .$

Step by step solution

01

Step 1. Given Information.

The given function and the interval is $f (x) = \sin x, [0, 2 蟺] .$

02

Step 2. Approximating the average value.

To approximate the average value of f from x = a to x = b, using a sample size of at least eight, we will use the formula: $(f (x_{0}) + . . . . . . f (x_{n})) 鈻� x, 鈻� x = \frac{b - a}{n} .$

Thus,

$鈻� x = \frac{2 蟺 - 0}{8} 鈻� x = \frac{蟺}{4}$

Now,

role="math" localid="1648725812220" $= (f (\frac{蟺}{4}) + f (\frac{蟺}{2}) + f (\frac{3 蟺}{4}) + f (蟺) + f (\frac{5 蟺}{4}) + f (\frac{3 蟺}{2}) + f (\frac{7 蟺}{4}) + f (2 蟺)) (\frac{蟺}{4}) = (\sin (\frac{蟺}{4}) + \sin (\frac{蟺}{2}) + \sin (\frac{3 蟺}{4}) + \sin (蟺) + \sin (\frac{5 蟺}{4}) + \sin (\frac{3 蟺}{2}) + \sin (\frac{7 蟺}{4}) + \sin (2 蟺)) (\frac{蟺}{4}) = 0$

Therefore, the average value is $0 .$

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