use left - endpoint approximation to approximate the area under the curve f(x)=-0.3x² + 22 between x = 1 and…

Answer:

1. For 6 rectangles: 102.3
2. For 12 rectangles: 104.475

Explanation:

Step1: Calculate the width of rectangles ($\Delta x$) for 6 - rectangles case

$\Delta x=\frac{b - a}{n}=\frac{7 - 1}{6}=1$

Step2: Determine the left - endpoints for 6 - rectangles case

The left - endpoints are $x_0 = 1,x_1=2,x_2 = 3,x_3=4,x_4 = 5,x_5=6$.

Step3: Calculate the sum for 6 - rectangles case

$A_1=\sum_{i = 0}^{5}f(x_i)\Delta x$ $f(1)=-0.3\times1^2 + 22=21.7$ $f(2)=-0.3\times2^2 + 22=20.8$ $f(3)=-0.3\times3^2 + 22=19.3$ $f(4)=-0.3\times4^2 + 22=17.2$ $f(5)=-0.3\times5^2 + 22=14.5$ $f(6)=-0.3\times6^2 + 22=11.2$ $A_1=(21.7 + 20.8+19.3 + 17.2+14.5 + 11.2)\times1=102.3$

Step4: Calculate the width of rectangles ($\Delta x$) for 12 - rectangles case

$\Delta x=\frac{b - a}{n}=\frac{7 - 1}{12}=0.5$

Step5: Determine the left - endpoints for 12 - rectangles case

The left - endpoints are $x_0 = 1,x_1 = 1.5,x_2=2,x_3 = 2.5,x_4=3,x_5 = 3.5,x_6=4,x_7 = 4.5,x_8=5,x_9 = 5.5,x_{10}=6,x_{11}=6.5$.

Step6: Calculate the sum for 12 - rectangles case

$A_2=\sum_{i = 0}^{11}f(x_i)\Delta x$ $f(1)=21.7$ $f(1.5)=-0.3\times(1.5)^2 + 22=21.325$ $f(2)=20.8$ $f(2.5)=-0.3\times(2.5)^2 + 22=20.125$ $f(3)=19.3$ $f(3.5)=-0.3\times(3.5)^2 + 22=18.325$ $f(4)=17.2$ $f(4.5)=-0.3\times(4.5)^2 + 22=15.925$ $f(5)=14.5$ $f(5.5)=-0.3\times(5.5)^2 + 22=12.825$ $f(6)=11.2$ $f(6.5)=-0.3\times(6.5)^2 + 22=8.225$ $A_2=(21.7+21.325 + 20.8+20.125+19.3+18.325+17.2+15.925+14.5+12.825+11.2+8.225)\times0.5 = 104.475$