$3$

$6.(4$ pt$.)$ We defined the area $A$ of the region $S$ that lies under the graph of the continuous function $f$ as the limit of the sum of the areas of the approximating rectangles:

$A=\underset{n}{lim}{R}_n=\underset{n}{lim}[f(x_1)xf(x_2)x$ cdots $f(x_n)x]=\underset{n}{lim}{\displaystyle \underset{i=1}{\overset{n}{}}}f(x_i)x$

Use this definition to find an expression for the area under the graph of $f$ as a limit$.$ Do not evaluate the limit$.$

$f(x)=ta{n}^{-1}(3x-1)on0,1$

$(4$ pt$.)$ Consider the following limit of a Riemann sum for a function $f$ on $a,b.$ Identify $f$ and express the limit as a definite integral. Do not evaluate the integral.

$\underset{n}{lim}{\displaystyle \underset{i=1}{\overset{n}{}}}|x_i^2-1|xon-2,2$

$f$ ; Definite integral: $$

$8.(4$ pt$.)$ If $f(x)$ is the slope of a trail at a distance of $x$ miles form the start of the trail, what does ${\displaystyle \underset{3}{\overset{5}{}}}f(x)dx$ represent? Write a complete grammatically correct sentence.

$9.(10$ pt $.)$ The acceleration function $($in $\frac{m}{s^2})$ and the initial velocity are given for a particle moving along a line. Find $($a$)$ the velocity at time $t$ and $($b$)$ the total distance traveled during the given time interval. Write your final answer rounded to hundredths.

$a(t)=2t3,v(0)=-4,0t3$