To estimate the area under $f(x) = 4 - x^2$ between $x = -2$ and $x = 2$, we can use finite approximations based on lower and upper sums. Let's break this into the required parts:

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1. Preliminary Details:

• Interval: $[-2, 2]$.
• Width of each rectangle ($\Delta x$):
  • For 2 rectangles: $\Delta x = \frac{2 - (-2)}{2} = 2$.
  • For 4 rectangles: $\Delta x = \frac{2 - (-2)}{4} = 1$.
• Heights of the rectangles depend on the function $f(x) = 4 - x^2$ at specific points within each subinterval:
  • For lower sum: Use the minimum value of $f(x)$ in each subinterval.
  • For upper sum: Use the maximum value of $f(x)$ in each subinterval.

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a. Lower Sum with 2 Rectangles

Each rectangle has width $\Delta x = 2$, and the subintervals are:

1. $[-2, 0]$: Minimum height is $f(0) = 4$.
2. $[0, 2]$: Minimum height is $f(2) = 0$.

The lower sum is: [ \text{Lower Sum} = \Delta x \cdot \left[\min f(x \text{ in } [-2, 0]) + \min f(x \text{ in } [0, 2])\right] = 2 \cdot (0 + 0) = 0. ]