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 ≈ $square$. (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 2 rectangles, width $\Delta x = \frac{8}{2} = 4$.

Step2: Find midpoints of each subinterval

First subinterval: $[-4, 0]$, midpoint $x_1 = \frac{-4+0}{2} = -2$.
Second subinterval: $[0, 4]$, midpoint $x_2 = \frac{0+4}{2} = 2$.

Step3: Compute function values at midpoints

For $x_1=-2$: $y_1 = 16 - (-2)^2 = 16 - 4 = 12$.
For $x_2=2$: $y_2 = 16 - (2)^2 = 16 - 4 = 12$.

Step4: Calculate total estimated area

Area = $\Delta x \times (y_1 + y_2) = 4 \times (12 + 12) = 4 \times 24$

Answer:

96