Question

question given the graph of the function f(x) below, use a left riemann sum with 3 rectangles to approximate the integral ∫₅³ f(x) dx. give your answer rounded to the nearest tenth, if necessary. provide your answer below.

Explanation:

Step1: Determine the width of sub - intervals

The interval is from \(x = 3\) to \(x=6\). We want \(n = 3\) rectangles. The width of each sub - interval \(\Delta x=\frac{b - a}{n}\), where \(a = 3\), \(b = 6\) and \(n=3\). So \(\Delta x=\frac{6 - 3}{3}=1\).

Step2: Identify the left - hand endpoints

The left - hand endpoints of the three sub - intervals \([3,4]\), \([4,5]\), \([5,6]\) are \(x_1 = 3\), \(x_2 = 4\), \(x_3 = 5\).

Step3: Estimate the function values at the left - hand endpoints

From the graph, \(f(3)\approx3\), \(f(4)\approx4\), \(f(5)\approx5\).

Step4: Calculate the left - Riemann sum

The left - Riemann sum \(L_3=\sum_{i = 1}^{3}f(x_i)\Delta x\). Substituting the values, we get \(L_3=f(3)\times1+f(4)\times1+f(5)\times1=(3 + 4+5)\times1 = 12\).

Answer:

\(12\)