find the total area of the rectangles in the figure below, where the equation of the curve is f(x)= -x² + 14.

Explanation:

Step1: Determine width and height of each rectangle

Assume each grid - square has side - length 1. The width of each rectangle is 1. For the left - most rectangle (at (x=-2)), (f(-2)=-(-2)^{2}+14=-4 + 14 = 10). For the second rectangle (at (x=-1)), (f(-1)=-(-1)^{2}+14=-1 + 14 = 13). For the third rectangle (at (x = 1)), (f(1)=-(1)^{2}+14=-1 + 14 = 13). For the right - most rectangle (at (x = 2)), (f(2)=-(2)^{2}+14=-4 + 14 = 10).

Step2: Calculate area of each rectangle

The area of a rectangle is (A = \text{width}\times\text{height}). Since the width of each rectangle is 1, the areas of the rectangles are equal to their heights. The areas of the rectangles are (A_1 = 10), (A_2 = 13), (A_3 = 13), (A_4 = 10).

Step3: Calculate total area

The total area (A_{total}=A_1 + A_2+A_3 + A_4). (A_{total}=10 + 13+13 + 10=46).

Answer:

46