find the total area of the rectangles in the figure below, where the equation of the line is f(x) = 3x - 9.

Answer

Explanation:

Step1: Determine width and heights

The width of each rectangle is $\Delta x = 1$. The $x$-values for the rectangles are $x = 4,5,6,7$. Calculate the heights using $f(x)=3x - 9$.

Step2: Calculate height at $x = 4$

$f(4)=3\times4 - 9=12 - 9 = 3$.

Step3: Calculate height at $x = 5$

$f(5)=3\times5 - 9=15 - 9 = 6$.

Step4: Calculate height at $x = 6$

$f(6)=3\times6 - 9=18 - 9 = 9$.

Step5: Calculate height at $x = 7$

$f(7)=3\times7 - 9=21 - 9 = 12$.

Step6: Calculate total area

The area of a rectangle is $A = \text{width}\times\text{height}$. The total area $A_{total}=\sum_{i = 1}^{4}f(x_i)\Delta x$. Since $\Delta x = 1$, $A_{total}=f(4)+f(5)+f(6)+f(7)=3 + 6+9 + 12=30$.