Answer
$1$
Work Step by Step
$ƒ(t) = \frac{1}{2}+\sin^{2}(\pi t)$ on the interval [0, 2] using a finite sum, we can partition the interval into four subintervals of equal length and evaluate ƒ at the subinterval midpoints.
The length of each subinterval can be calculated as follows:
Length of subinterval = $\frac{ (\text{upper bound - lower bound})}{\text{number of subintervals}}$
= $\frac{(2 - 0)}{4}$
= 0.5
Now, we can calculate the subinterval midpoints:
Subinterval 1: midpoint = $\frac{(0 + 0.5)}{2}$= 0.25
Subinterval 2: midpoint = $\frac{(0.5 + 1)}{2}$= 0.75
Subinterval 3: midpoint = $\frac{ (1 + 1.5)}{2}$= 1.25
Subinterval 4: midpoint = $\frac{ (1.5 + 2)}{2}$= 1.75
Next, we evaluate ƒ at these subinterval midpoints:
ƒ(0.25) = $\frac{1}{2}$+$\sin^{2}(\pi(0.25))$ = 0.5 + 0.5 = 1
ƒ(0.75) = $\frac{1}{2}$+$\sin^{2}(\pi(0.75))$ = 0.5 + 0.5 = 1
ƒ(1.25) = $\frac{1}{2}$+$\sin^{2}(\pi(1.25))$ = 0.5 + 0.5 = 1
ƒ(1.75) = $\frac{1}{2}$+$\sin^{2}(\pi(1.75))$ = 0.5 + 0.5 = 1
Finally, we can calculate the average value of ƒ on the interval [0, 2] using the finite sum:
Average value of ƒ = $\frac{(\text{sum of function values})}{ \text{number of subintervals}}$
= $\frac{(1 + 1 + 1 + 1)}{4}$
= 1
