1. Let $$ f(x)=4-2 x $$. (a) Sketch the region $$ R $$ under the graph of $$ f $$ on the interval $$ [0,2] $$ and find its exact area using geometry. (b) Use a Riemann sum with five subintervals of equal length (i.e., $$ n=5 $$ ) to ap- proximate the area of $$ R $$. Choose the representative points to be the left end- points of the subintervals. (c) Repeat part (b) with ten subintervals of equal length (i.e., $$ n=10 $$ ). (d) Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger $$ n $$ ? 2. Repeat Exercises 1 (b) and 1 (c), choosing the representative points to be the right endpoints of the subintervals. 3. Repeat Exercises 1 (b) and 1 (c), choosing the representative points to be the mid- points of the subintervals.

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Graham Pritchard · Accepted Answer

(a) The region under the graph of $$ f(x) = 4 - 2x $$ on the interval $$ [0,2] $$ is a triangle with vertices at $$ (0,4) $$, $$ (2,0) $$, and the origin. The exact area is 4 square units. (b) The Riemann sum approximation using five left end-points is 0. (c) The Riemann sum approximation using ten left end-points is also 0.

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