Question

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approximate the area between f(x) and the x-axis from x = -3 to x = 3 using a midpoint riemann sum with 3 equal subdivisions.
m(3) =
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Explanation:

Step1: Find width of each subinterval

The interval is from $x=-3$ to $x=3$, so total length is $3 - (-3) = 6$. Divide by 3 subdivisions:
$\Delta x = \frac{3 - (-3)}{3} = 2$

Step2: Identify subintervals and midpoints

Subintervals: $[-3,-1]$, $[-1,1]$, $[1,3]$
Midpoints: $x_1=-2$, $x_2=0$, $x_3=2$

Step3: Read $f(x)$ at midpoints from graph

From the graph:
$f(-2)=6$, $f(0)=3$, $f(2)=5$

Step4: Calculate midpoint Riemann sum

Sum $\Delta x \cdot [f(x_1)+f(x_2)+f(x_3)]$:
$M(3) = 2 \cdot (6 + 3 + 5)$

Step5: Compute final value

$2 \cdot 14 = 28$

Answer:

28