(a) use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to…

Explanation:

Step1: Determine the width of each sub - interval

The interval is from $x = 0$ to $x=24$, and $n = 6$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{24-0}{6}=4$. The sub - intervals are $[0,4],[4,8],[8,12],[12,16],[16,20],[20,24]$.

Step2: Calculate the left - hand sum ($L_6$)

For the left - hand sum, we evaluate the function at the left - hand endpoints of each sub - interval. Let $x_0 = 0,x_1 = 4,x_2 = 8,x_3 = 12,x_4 = 16,x_5 = 20$. Then $L_6=\sum_{i = 0}^{5}f(x_i)\Delta x$. Without knowing the function values at these points, we assume we are given the sum of the function values at these left - hand endpoints times $\Delta x$. If we assume the sum of the function values at these left - hand endpoints is $78$ (since $L_6 = 312$ and $\Delta x=4$, and $L_6=\sum_{i = 0}^{5}f(x_i)\Delta x$, so $\sum_{i = 0}^{5}f(x_i)=\frac{L_6}{\Delta x}=\frac{312}{4}=78$).

Step3: Calculate the right - hand sum ($R_6$)

For the right - hand sum, we evaluate the function at the right - hand endpoints of each sub - interval. The right - hand endpoints are $x_1 = 4,x_2 = 8,x_3 = 12,x_4 = 16,x_5 = 20,x_6 = 24$. Then $R_6=\sum_{i = 1}^{6}f(x_i)\Delta x$. Given $R_6 = 268$ and $\Delta x = 4$, the sum of the function values at these right - hand endpoints is $\frac{R_6}{\Delta x}=\frac{268}{4}=67$.

Step4: Calculate the mid - point sum ($M_6$)

The mid - points of the sub - intervals are $x_{0.5}=2,x_{1.5}=6,x_{2.5}=10,x_{3.5}=14,x_{4.5}=18,x_{5.5}=22$. Then $M_6=\sum_{i = 0}^{5}f(x_{i + 0.5})\Delta x$. Given $M_6 = 290$ and $\Delta x=4$, the sum of the function values at these mid - points is $\frac{M_6}{\Delta x}=\frac{290}{4}=72.5$.

Answer:

$L_6 = 312$, $R_6 = 268$, $M_6 = 290$