Question

the continuous function f is known to be increasing for all x. selected values of f are given in the table above. let l be the left - riemann sum approximation for $int_{2}^{11}f(x)dx$ using the four subintervals indicated by the table. which of the following statements is true? a $l = 13.95$ and is an underestimate for $int_{2}^{11}f(x)dx$. b $l = 13.95$ and is an overestimate for $int_{2}^{11}f(x)dx$. c $l = 15.5$ and is an underestimate for $int_{2}^{11}f(x)dx$. d $l = 15.5$ and is an overestimate for $int_{2}^{11}f(x)dx$.

Explanation:

Step1: Recall left - Riemann sum formula

The left - Riemann sum $L$ for $\int_{a}^{b}f(x)dx$ with subintervals $[x_0,x_1],[x_1,x_2],\cdots,[x_{n - 1},x_n]$ is $L=\sum_{i = 0}^{n-1}f(x_i)\Delta x_i$, where $a=x_0$ and $b = x_n$. Here, $a = 2$, $b=11$, and the subintervals are $[2,4],[4,6],[6,7],[7,11]$. So, $\Delta x_1=4 - 2=2$, $\Delta x_2=6 - 4 = 2$, $\Delta x_3=7 - 6=1$, $\Delta x_4=11 - 7 = 4$.

Step2: Calculate the left - Riemann sum

$L=f(2)\times(4 - 2)+f(4)\times(6 - 4)+f(6)\times(7 - 6)+f(7)\times(11 - 7)$. Substitute $f(2)=0.7$, $f(4)=1.1$, $f(6)=1.9$, $f(7)=2.5$ into the formula: $L=0.7\times2+1.1\times2 + 1.9\times1+2.5\times4=1.4 + 2.2+1.9 + 10=15.5$.

Step3: Determine if it's an over - or under - estimate

Since the function $f(x)$ is increasing for all $x$, when we use a left - Riemann sum to approximate $\int_{2}^{11}f(x)dx$, the left - Riemann sum is an underestimate. Because for an increasing function, the left - hand endpoint of each subinterval gives the smallest value of the function on that subinterval.

Answer:

C. $L = 15.5$ and is an underestimate for $\int_{2}^{11}f(x)dx$.