given the graph of the function f(x) below, use a left riemann sum with 5 rectangles to approximate the…
given the graph of the function f(x) below, use a left riemann sum with 5 rectangles to approximate the integral ∫₁⁶ f(x) dx.
Answer
Explanation:
Step1: Calculate width of rectangles
The interval is from $a = 1$ to $b=6$, and $n = 5$. The width $\Delta x=\frac{b - a}{n}=\frac{6 - 1}{5}=1$.
Step2: Identify left - hand endpoints
The left - hand endpoints of the 5 sub - intervals $[1,2],[2,3],[3,4],[4,5],[5,6]$ are $x_1 = 1,x_2 = 2,x_3 = 3,x_4 = 4,x_5 = 5$.
Step3: Determine function values at endpoints
From the graph, $f(x_1)=9,f(x_2)=7,f(x_3)=7,f(x_4)=6,f(x_5)=7$.
Step4: Calculate left Riemann sum
The left Riemann sum $L_5=\sum_{i = 1}^{5}f(x_i)\Delta x$. Substitute the values: $L_5=(9 + 7+7 + 6+7)\times1$. $L_5=9 + 7+7 + 6+7=36$.
Answer:
$36$