consider the definite integral ∫₀² x³ dx\n a. using four rectangles, find the left - hand riemann sum for…

Explanation:

Step1: Calculate the width of each sub - interval

The interval is $[a,b]=[0,2]$ and $n = 4$. The width $\Delta x=\frac{b - a}{n}=\frac{2-0}{4}=0.5$.

Step2: Find the left - hand Riemann sum

The left - hand endpoints are $x_0 = 0,x_1=0.5,x_2 = 1,x_3=1.5$. The left - hand Riemann sum $L_4=\sum_{i = 0}^{3}f(x_i)\Delta x$. $f(x)=x^3$, so $L_4=f(0)\times0.5+f(0.5)\times0.5+f(1)\times0.5+f(1.5)\times0.5$. $f(0)=0^3 = 0$, $f(0.5)=(0.5)^3=0.125$, $f(1)=1^3 = 1$, $f(1.5)=(1.5)^3=3.375$. $L_4=(0 + 0.125+1 + 3.375)\times0.5=(4.5)\times0.5 = 2.25$.

Step3: Find the right - hand Riemann sum

The right - hand endpoints are $x_1 = 0.5,x_2=1,x_3 = 1.5,x_4=2$. The right - hand Riemann sum $R_4=\sum_{i = 1}^{4}f(x_i)\Delta x$. $f(0.5)=(0.5)^3=0.125$, $f(1)=1^3 = 1$, $f(1.5)=(1.5)^3=3.375$, $f(2)=2^3 = 8$. $R_4=(0.125 + 1+3.375+8)\times0.5=(12.5)\times0.5=6.25$.

Step4: Calculate the average of the estimates

The average of $L_4$ and $R_4$ is $\frac{L_4+R_4}{2}=\frac{2.25 + 6.25}{2}=\frac{8.5}{2}=4.25$.

Answer:

a. $2.25$ b. $6.25$ c. $4.25$