Question

let ( f(x) ) be a function with the following table of values:

\begin{array}{c|ccccccc} x = & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ f(x) = & 3 & -8 & -5 & -5 & 6 & -2 & -7 end{array}

evaluate the riemann sum of ( f(x) ) on the interval ( 1,7 ) using three equal sub - intervals and left endpoints:

. (your answer should be a number.)

question 5 (1 point)

consider a function ( f ) with the following table of values:

\begin{array}{|c|c|c|c|c|c|c|c|} hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ hline f(x) & 4 & -2 & -3 & 0 & 1 & -1 & 2 \\ hline end{array}

approximate ( int_{2}^{6}f(x)dx ) using a riemann sum with two rectangles of equal width and midpoints.

Explanation:

Step1: Calculate the width of sub - intervals

For the first problem:
The interval is \([1,7]\), \(n = 3\). The width \(\Delta x=\frac{7 - 1}{3}=2\).
For the second problem:
The interval is \([2,6]\), \(n = 2\). The width \(\Delta x=\frac{6 - 2}{2}=2\).

Step2: Find the left - endpoints (first problem) or mid - points (second problem)

First problem (left - endpoints):
The sub - intervals are \([1,3]\), \([3,5]\), \([5,7]\). The left - endpoints are \(x_1 = 1\), \(x_2=3\), \(x_3 = 5\).
\(f(x_1)=3\), \(f(x_2)=- 5\), \(f(x_3)=6\).
The Riemann sum \(S=\sum_{i = 1}^{3}f(x_i)\Delta x=(3-5 + 6)\times2=(4)\times2 = 8\).
Second problem (mid - points):
The sub - intervals are \([2,4]\), \([4,6]\). The mid - points are \(x_1=\frac{2 + 4}{2}=3\), \(x_2=\frac{4+6}{2}=5\).
\(f(x_1)=0\), \(f(x_2)=-1\).
The Riemann sum \(S=\sum_{i = 1}^{2}f(x_i)\Delta x=(0-1)\times2=-2\).

Answer:

For the first problem: \(8\)
For the second problem: \(-2\)