Use finite approximations to estimate the area under the graph of the function $f(x)=4-x^{2}$ between $x=-2$ and $x=2$ for each of the following cases.
a. Using a lower sum with two rectangles of equal width
b. Using a lower sum with four rectangles of equal width
c. Using an upper sum with two rectangles of equal width
d. Using an upper sum with four rectangles of equal width

Step 1 ：We are given the function \(f(x)=4-x^{2}\) and we are asked to estimate the area under the curve of this function between \(x=-2\) and \(x=2\) using a lower sum with two rectangles of equal width.

Step 2 ：The width of each rectangle will be the total width of the interval divided by the number of rectangles. In this case, the total width is \(2 - (-2) = 4\), so the width of each rectangle will be \(4/2 = 2\).

Step 3 ：The height of each rectangle will be the minimum value of the function on the interval that the rectangle covers. Since the function \(f(x)=4-x^{2}\) is symmetric about the y-axis, the minimum value on each interval will be the value of the function at the right endpoint of the interval.

Step 4 ：So, the area of each rectangle will be width times height, and the total area will be the sum of the areas of the two rectangles.

Step 5 ：Let's calculate this. The first rectangle covers the interval from \(x=-2\) to \(x=0\), and the second rectangle covers the interval from \(x=0\) to \(x=2\). The height of the first rectangle is \(f(-2)=4-(-2)^{2}=0\), and the height of the second rectangle is \(f(0)=4-0^{2}=4\).

Step 6 ：So, the area of the first rectangle is \(2*0=0\), and the area of the second rectangle is \(2*4=8\).

Step 7 ：The total area under the curve of the function \(f(x)=4-x^{2}\) between \(x=-2\) and \(x=2\) using a lower sum with two rectangles of equal width is \(0+8=8\).

Step 8 ：Final Answer: The area under the curve of the function \(f(x)=4-x^{2}\) between \(x=-2\) and \(x=2\) using a lower sum with two rectangles of equal width is \(\boxed{8}\).