Calculate the indicated Riemann sum $S_{3}$ for the function $f(x)=x^{2}-5 x-24$. Partition $[0,6]$ into three subintervals of equal length, and let $c_{1}=0.5, c_{2}=3.1$, and $c_{3}=4.9$.
Answer
Final Answer: The Riemann sum \(S_{3}\) for the function \(f(x)=x^{2}-5 x-24\) over the interval \([0,6]\) with the points \(c_{1}=0.5, c_{2}=3.1\), and \(c_{3}=4.9\) is \(\boxed{-161.26}\).
Step 1 ：We are given the function \(f(x)=x^{2}-5 x-24\), the interval \([0,6]\), and the points \(c_{1}=0.5, c_{2}=3.1\), and \(c_{3}=4.9\).
Step 2 ：The interval \([0,6]\) is partitioned into three subintervals of equal length, so \(\Delta x_{i} = \frac{6-0}{3} = 2\) for all \(i\).
Step 3 ：The Riemann sum \(S_{n}\) is defined as: \(S_{n} = \sum_{i=1}^{n} f(c_{i}) \Delta x_{i}\) where \(\Delta x_{i}\) is the width of the subinterval \([x_{i-1}, x_{i}]\), and \(c_{i}\) is a point in the subinterval \([x_{i-1}, x_{i}]\).
Step 4 ：We can now substitute these values into the formula for the Riemann sum to find \(S_{3}\).
Step 5 ：The Riemann sum \(S_{3}\) for the function \(f(x)=x^{2}-5 x-24\) over the interval \([0,6]\) with the points \(c_{1}=0.5, c_{2}=3.1\), and \(c_{3}=4.9\) is calculated to be -161.26.
Step 6 ：Final Answer: The Riemann sum \(S_{3}\) for the function \(f(x)=x^{2}-5 x-24\) over the interval \([0,6]\) with the points \(c_{1}=0.5, c_{2}=3.1\), and \(c_{3}=4.9\) is \(\boxed{-161.26}\).
