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Consider the area, \(\mathcal{A}\), under the curve of the function \(f(x)=\sqrt{x}\) from \(x=1\) to \(x=3\).
  1. Divide the interval into 4 subintervals of equal width. Then, estimate the area by summing the areas of rectangles whose heights are determined by the function's value at:
    1. the left-hand endpoint of each subinterval (\(L_4\)).
    2. the right-hand endpoint of each subinterval (\(R_4\)).
  2. By observing the function's behavior, state whether your estimations are overestimates or underestimates, and write an inequality that relates \(L_4\), \(R_4\), and the true area \(\mathcal{A}\).

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