approximate the area under the following curve and above the x - axis on the given interval, using…

approximate the area under the following curve and above the x - axis on the given interval, using rectangles whose height is the value of the function at the left side of the rectangle. (a) use two rectangles. (b) use four rectangles. (c) use a graphing calculator (or other technology) and 40 rectangles. f(x)=5 + x²;-2,2 (a) the approximated area when using two rectangles is 28 square units. (type an integer or decimal rounded to two decimal places as needed.) (b) the approximated area when using four rectangles is 26 square units. (type an integer or decimal rounded to two decimal places as needed.) (c) the approximated area when using 40 rectangles is square units. (type an integer or decimal rounded to two decimal places as needed.)

approximate the area under the following curve and above the x - axis on the given interval, using rectangles whose height is the value of the function at the left side of the rectangle. (a) use two rectangles. (b) use four rectangles. (c) use a graphing calculator (or other technology) and 40 rectangles. f(x)=5 + x²;-2,2 (a) the approximated area when using two rectangles is 28 square units. (type an integer or decimal rounded to two decimal places as needed.) (b) the approximated area when using four rectangles is 26 square units. (type an integer or decimal rounded to two decimal places as needed.) (c) the approximated area when using 40 rectangles is square units. (type an integer or decimal rounded to two decimal places as needed.)

Answer

Explanation:

Step1: Calculate the width of each sub - interval

The interval is $[a,b]=[-2,2]$. The width of a sub - interval $\Delta x=\frac{b - a}{n}$, where $n$ is the number of rectangles. For $n = 40$, $\Delta x=\frac{2-(-2)}{40}=\frac{4}{40}=0.1$.

Step2: Determine the left - hand endpoints

The left - hand endpoints $x_i=a + i\Delta x$ for $i = 0,1,\cdots,n - 1$. Here, $x_i=-2+0.1i$.

Step3: Calculate the sum of the areas of the rectangles

The area of each rectangle is $A_i=f(x_i)\Delta x$, where $f(x)=5 + x^{2}$. The sum $S=\sum_{i = 0}^{39}f(x_i)\Delta x=\sum_{i = 0}^{39}(5+( - 2+0.1i)^{2})\times0.1$. First, expand $( - 2+0.1i)^{2}=4 - 0.4i+0.01i^{2}$. Then $f(x_i)=5+4 - 0.4i+0.01i^{2}=9 - 0.4i+0.01i^{2}$. $S = 0.1\sum_{i = 0}^{39}(9 - 0.4i+0.01i^{2})$. We know that $\sum_{i = 0}^{n}c=cn$ (where $c$ is a constant), $\sum_{i = 0}^{n}i=\frac{n(n + 1)}{2}$, and $\sum_{i = 0}^{n}i^{2}=\frac{n(n + 1)(2n+1)}{6}$. $\sum_{i = 0}^{39}9=9\times40 = 360$, $\sum_{i = 0}^{39}(-0.4i)=-0.4\times\frac{39\times40}{2}=-0.4\times780=-312$, $\sum_{i = 0}^{39}(0.01i^{2})=0.01\times\frac{39\times40\times79}{6}=0.01\times20280 = 202.8$. $S=0.1\times(360-312 + 202.8)=0.1\times250.8 = 25.08$.

Answer:

$25.08$