1. approximate the area under the curve f(x) = x² + 1 and above the x - axis on the interval 2,8: a) using…

1. approximate the area under the curve f(x) = x² + 1 and above the x - axis on the interval 2,8: a) using rectangles whose height is the value of the function is the midpoint of the subinterval using 3 rectangles

1. approximate the area under the curve f(x) = x² + 1 and above the x - axis on the interval 2,8: a) using rectangles whose height is the value of the function is the midpoint of the subinterval using 3 rectangles

Answer

Explanation:

Step1: Calculate width of sub - intervals

The interval is $[a,b]=[2,8]$ and $n = 3$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{8 - 2}{3}=2$.

Step2: Determine sub - intervals and midpoints

The sub - intervals are $[2,4]$, $[4,6]$, $[6,8]$. The midpoints are $x_1 = 3$, $x_2=5$, $x_3 = 7$.

Step3: Calculate the value of the function at midpoints

$f(x)=x^{2}+1$. $f(x_1)=3^{2}+1=9 + 1=10$. $f(x_2)=5^{2}+1=25 + 1=26$. $f(x_3)=7^{2}+1=49+1=50$.

Step4: Calculate the area approximation

The area $A\approx\sum_{i = 1}^{3}f(x_i)\Delta x$. $A\approx f(x_1)\Delta x+f(x_2)\Delta x+f(x_3)\Delta x$. $A\approx(10 + 26+50)\times2=(86)\times2 = 172$.

Answer:

172