Answer
$6$
Work Step by Step
The interval $[a,b]$ is subdivided into $n$ intervals, and the function
$f(x)$ is a continuous function over $[a,b]$.
Here, Left Riemann sum $=\displaystyle \sum_{k=0}^{n-1}f(x_{k})\Delta x$
$=[f(x_{0})+f(x_{1})+\cdots+f(x_{n-1})]\Delta x$
where $a=x_{0} < x_{1} < \cdots < x_{n}=b$ are the endpoints of the subdivisions and $\displaystyle \Delta x=\frac{b-a}{n}$
Here, we have $ [a,b]=[0,8]$ and $ n=4$
$\displaystyle \Delta x=\frac{b-a}{n}=\frac{8-0}{4}=2$
Now,
$\text{Left Riemann sum}=(1+1+1.5+2) =5.5$ and $\text{Right Riemann sum}=(1+1.5+2+2) =6.5$
This gives: $Average=\dfrac{5.5+6.5}{2}=6$
Therefore, the total change of $f(t)$ $=\int_1^5 f'(t) \ dt=6$
