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

## Discussion

## Video Transcript

to solve this problem, we must first calculate Delta acts, which is pi minus zero. Divide by on which gives us pi over end and simplest form, which means now we can plug in the formula. We know we have exit by which is pi I over and and Delta X purity establishes pi over an which means we have the limit is un approaches Infinity square of sign of this value over here, Axl by times pi over end which is our delta X.

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