Question

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.
$y = 16 - x^2$ between $x = -4$ and $x = 4$
for two rectangles, area ≈ 96. (type an integer or a decimal.)
for four rectangles, area ≈ $\square$. (type an integer or a decimal.)

Explanation:

Step1: Calculate width of each rectangle

The interval is from $x=-4$ to $x=4$, so total length is $4 - (-4) = 8$. For 4 rectangles, width $\Delta x = \frac{8}{4} = 2$.

Step2: Find midpoints of each subinterval

Subintervals: $[-4,-2], [-2,0], [0,2], [2,4]$. Midpoints: $x_1=-3$, $x_2=-1$, $x_3=1$, $x_4=3$.

Step3: Calculate height at each midpoint

For $x=-3$: $y=16 - (-3)^2 = 16-9=7$
For $x=-1$: $y=16 - (-1)^2 = 16-1=15$
For $x=1$: $y=16 - (1)^2 = 16-1=15$
For $x=3$: $y=16 - (3)^2 = 16-9=7$

Step4: Sum areas of all rectangles

Area = $\Delta x \times (y_1 + y_2 + y_3 + y_4) = 2 \times (7+15+15+7)$
$=2 \times 44 = 88$

Answer:

88