Question: Approximate the area of the region beneath the graph of \(f(x)=e^{-x^{2}}\) from \(x=-1\) to \(x=1\) using four left rectangles, four right rectangles, and four midpoint rectangles.

1. In each case,

1. sketch the graph of \(f\) from \(x=-1\) to \(x=1\).
2. label the points on the \(x\) -axis, and draw the rectangles.
3. calculate the approximate areas.

b. Proceed as in part \(a\) to approximate the area of the region beneath the graph of \(f\) from \(x=-1\) to \(x=1\) using eight left rectangles, eight right rectangles, and eight midpoint rectangles.

c. The actual area, to nine decimal places, of the region beneath the graph of \(f(x)=e^{-x^{2}}\) is 1.493648266. Which of the approximations found in part \(b\) is the most accurate?

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