Question

question 5 (1 point)
consider a function ( f ) with the following table of values:

approximate ( int_{0}^{6} f(x) d x ) using a riemann sum with three rectangles of equal width and midpoints.
-6
1
-3
3
4

Explanation:

Step1: Calculate the width of each rectangle

The interval is from \(a = 0\) to \(b=6\), and \(n = 3\). The width \(\Delta x=\frac{b - a}{n}=\frac{6-0}{3}=2\).

Step2: Determine the mid - points

The sub - intervals are \([0,2]\), \([2,4]\), \([4,6]\). The mid - points are \(x_1 = 1\), \(x_2=3\), \(x_3 = 5\).

Step3: Calculate the Riemann sum

The Riemann sum \(M_n=\sum_{i = 1}^{n}f(x_i)\Delta x\).
We know \(f(1)=-2\), \(f(3)=0\), \(f(5)=-1\) and \(\Delta x = 2\).

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Answer:

\(-6\)