Question 1206935
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The function f(x)=x^2 is monotonically increasing on the interval [0,2], so the heights of all of the upper rectangles will be the function values at the upper ends of the intervals.<br>
With 8 subintervals on [0,2], the width of each rectangle will be 2/8 = 1/4.<br>
The upper endpoints of the 8 intervals are 1/4, 2/4, 3/4, 4/4, 5/4, 6/4, 7/4, and 8/4.<br>
The function values at those endpoints -- i.e., the heights of the rectangles -- are 1/16, 4/16, 9/16, 16/16, 25/16, 36/16, 49/16, and 64/16.<br>
The sum of the areas of the eight rectangles is<br>
{{{(1/4)(1/16+4/16+9/16+16/16+25/16+36/16+49/16+64/16)=(1/64)(1+4+9+16+25+36+49+64)=(1/64)(204)=204/64=51/16}}}<br>
ANSWER: 51/16<br>