Problem 27 Medium Difficulty

Let $ A $ be the area under the graph of an increasing continuous function $ f $ from $ a $ to $ b $, and let $ L_n $ and $ R_n $ be the approximations to $ A $ with $ n $ subintervals using left and right endpoints, respectively.

(a) How are $ A $, $ L_n $, and $ R_n $ related?

(b) Show that $$ R_n - L_n = \frac{b - a}{n} [f(b) - f(a)] $$
Then draw a diagram to illustrate this equation by showing that the $ n $ rectangles representing $ R_n - L_n $ can be reassembled to form a single rectangle whose area is the right side of the equation.

(c) Deduce that $$ R_n - A < \frac{b - a}{n} [f(b) - f(a)] $$

Answer

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Video Transcript

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