Rectangular Estimations <title></head> <body bgcolor="#FFFFFF"> <center> <font size=5><b> Rectangular Estimations</b></font> </center><font size=3> 1. A table of values is given for f(x). Estimate the integral from -45 to 75 of f(x) using the methods specified below. <P><center> <table width=70% border=3><tr> <th>x <th>-45 <th>-30 <th>-15 <th>0 <th>15 <th>30 <th>45 <th>60 <th>75 <tr> <th>f(x) <th>-4 <th>1 <th>5 <th>12 <th>18 <th>25 <th>20 <th>8 <th>2 </table></center> (a) Use L<sub>4</sub> <br><br><Br><br>(b) Use R<sub>4</sub> <br><br><Br><br>(c) Use L<sub>8</sub> <br><br><Br><br>(d) Use R<sub>8</sub> <br><br><Br><br>(e) Use M<sub>2</sub> <br><br><Br><br>(f) Use M<sub>4</sub> <br><br><hr noshade><p> <P>2. (a) <font size=2>Let f(x) = x<sup>2</sup>. Estimate the integral from a=1 to b=4 of f(x) using L<sub>3</sub>, R<sub>3</sub>, L<sub>6</sub>, R<sub>6</sub>, L<sub>12</sub>, and R<sub>12</sub> (time-saving tip: once you find L<sub>n</sub>, you can find R<sub>n</sub> with only a little bit more arithmetic). (b) Which (if any0 of the above estimates are <i>definitely</i> over-estimates or under-estimates, and <i>why?</i>. (c) What is the average of L<sub>12</sub>, and R<sub>12</sub>? What do you think the actual integral might equal? </font></body>