using rectangles whose height is given by the value of the function at the midpoint of the rectangles base…

using rectangles whose height is given by the value of the function at the midpoint of the rectangles base (the midpoint rule), estimate the area under the graph of the following function, using two and then four rectangles. y = 1 - x² between x = - 1 and x = 1

using rectangles whose height is given by the value of the function at the midpoint of the rectangles base (the midpoint rule), estimate the area under the graph of the following function, using two and then four rectangles. y = 1 - x² between x = - 1 and x = 1

Answer

Explanation:

Step1: Calculate width for 2 - rectangles

The interval is $[a=-1,b = 1]$. For $n = 2$ rectangles, the width $\Delta x=\frac{b - a}{n}=\frac{1-(-1)}{2}=1$.

Step2: Find mid - points and heights for 2 - rectangles

The mid - points are $x_1=-0.5$ and $x_2 = 0.5$. The heights are $y_1=1-(-0.5)^2=1 - 0.25 = 0.75$ and $y_2=1-(0.5)^2=1 - 0.25 = 0.75$. The area $A_2=\sum_{i = 1}^{2}y_i\Delta x=(0.75 + 0.75)\times1 = 1.5$.

Step3: Calculate width for 4 - rectangles

For $n = 4$ rectangles, $\Delta x=\frac{b - a}{n}=\frac{1-(-1)}{4}=0.5$.

Step4: Find mid - points and heights for 4 - rectangles

The mid - points are $x_1=-0.75,x_2=-0.25,x_3 = 0.25,x_4 = 0.75$. The heights are $y_1=1-(-0.75)^2=1 - 0.5625 = 0.4375$, $y_2=1-(-0.25)^2=1 - 0.0625 = 0.9375$, $y_3=1-(0.25)^2=1 - 0.0625 = 0.9375$, $y_4=1-(0.75)^2=1 - 0.5625 = 0.4375$. The area $A_4=\sum_{i = 1}^{4}y_i\Delta x=(0.4375 + 0.9375+0.9375 + 0.4375)\times0.5=1.375$.

Answer:

The area using 2 rectangles is 1.5 and using 4 rectangles is 1.375.