To estimate the average value [tex]\( f_{\text{ave}} \)[/tex] of the function [tex]\( f \)[/tex] on the interval [tex]\([20, 50]\)[/tex] using the Midpoint Rule with three subintervals, follow these steps:
1. Divide the interval [tex]\([20, 50]\)[/tex] into three subintervals:
The interval from 20 to 50 is 30 units long. Dividing it into three subintervals, each subinterval has a width of:
[tex]\[
\text{Width} = \frac{50 - 20}{3} = 10
\][/tex]
2. Determine the midpoints of each subinterval:
The midpoints can be found by averaging the endpoints of each subinterval:
[tex]\[
\text{Midpoints} = \left(\frac{20 + 30}{2}, \frac{30 + 40}{2}, \frac{40 + 50}{2}\right) = (25, 35, 45)
\][/tex]
Alternatively, for better accuracy, we can use the given midpoint between values in pairs from the table:
[tex]\[
\text{Midpoints} = (22.5, 32.5, 42.5)
\][/tex]
3. Find the function values at these midpoints:
Using the table, take the [tex]\( f(x) \)[/tex] value for each midpoint. Here, the values are given as:
[tex]\[
f(22.5) = 37, \quad f(32.5) = 29, \quad f(42.5) = 49
\][/tex]
4. Calculate the sum of the function values at the midpoints:
[tex]\[
\text{Sum of function values} = 37 + 29 + 49 = 115
\][/tex]
5. Apply the Midpoint Rule to estimate the integral:
Multiply the sum of function values by the width of each subinterval:
[tex]\[
\text{Integral Estimate} = 10 \times 115 = 1150.0
\][/tex]
6. Calculate the average value of the function on the interval [tex]\([20, 50]\)[/tex]:
The average value [tex]\( f_{\text{ave}} \)[/tex] of [tex]\( f \)[/tex] is obtained by dividing the integral estimate by the length of the interval:
[tex]\[
f_{\text{ave}} = \frac{\text{Integral Estimate}}{50 - 20} = \frac{1150.0}{30} \approx 38.333333333333336
\][/tex]
So, the estimated average value [tex]\( f_{\text{ave}} \)[/tex] of [tex]\( f \)[/tex] on [tex]\([20, 50]\)[/tex] using the Midpoint Rule with three subintervals is:
[tex]\[
f_{\text{ave}} \approx 38.33
\][/tex]
