current attempt in progress use the figure below to estimate ∫₀²⁰ f(x) dx. this integral represents the area…
current attempt in progress use the figure below to estimate ∫₀²⁰ f(x) dx. this integral represents the area under the graph of f(x) from x = 0 to x = 20. the total area is approximately etextbook and media attempts: 0 of 4 used save for later
Answer
Explanation:
Step1: Divide the interval
Divide the interval $[0,20]$ into 5 sub - intervals of width $\Delta x=\frac{20 - 0}{5}=4$.
Step2: Use left - hand endpoints
Estimate the value of $f(x)$ at the left - hand endpoints of each sub - interval: $x_0 = 0,x_1=4,x_2 = 8,x_3=12,x_4 = 16$. From the graph, $f(0)\approx1,f(4)\approx2,f(8)\approx3,f(12)\approx3.5,f(16)\approx4$.
Step3: Apply the left - hand Riemann sum formula
The left - hand Riemann sum $L_5=\sum_{i = 0}^{4}f(x_i)\Delta x$. Substitute the values: $L_5=f(0)\times4+f(4)\times4+f(8)\times4+f(12)\times4+f(16)\times4$. [ \begin{align*} L_5&=(1\times4)+(2\times4)+(3\times4)+(3.5\times4)+(4\times4)\ &=4\times(1 + 2+3+3.5 + 4)\ &=4\times13.5\ &=54 \end{align*} ]
Answer:
54