find the total area of the rectangles in the figure below, where the equation of the curve is f(x)= -x² + 12.

Answer

Explanation:

Step1: Determine width of rectangles

The width of each rectangle is $\Delta x = 1$.

Step2: Find heights of rectangles

For $x=-2$, $f(-2)=-(-2)^2 + 12=8$. For $x = - 1$, $f(-1)=-(-1)^2+12 = 11$. For $x = 0$, $f(0)=-0^2 + 12=12$. For $x = 1$, $f(1)=-1^2+12 = 11$. For $x = 2$, $f(2)=-2^2+12 = 8$.

Step3: Calculate area of each rectangle

Area of rectangle $A_i=f(x_i)\Delta x$. $A_1 = 8\times1=8$, $A_2=11\times1 = 11$, $A_3=12\times1=12$, $A_4=11\times1 = 11$, $A_5=8\times1=8$.

Step4: Sum up areas of rectangles

$A=A_1 + A_2+A_3+A_4+A_5=8 + 11+12+11+8=50$.