Calculate the indicated Riemann sum $\mathrm{S}_{5}$, for the function $f(x)=15-3 x^{2}$. Partition $[-5,5]$ into five subintervals of equal length, and for each subinterval $\left[x_{k-1}, x_{k}\right]$, let $c_{k}=\left(x_{k-1}+x_{k}\right) / 2$.
\[
\mathrm{S}_{5}=\square
\]

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Answer

Step 4: Calculate the Riemann sum. The Riemann sum is -90.

Steps

Step 1 ：Step 1: Determine the length of each subinterval. The length is 2 units.

Step 2 ：Step 2: Determine the midpoints of each subinterval. The midpoints are -4, -2, 0, 2, and 4.

Step 3 ：Step 3: Evaluate the function at each midpoint. The function values are -33, 3, 15, 3, and -33.

Step 4 ：Step 4: Calculate the Riemann sum. The Riemann sum is -90.