- Calculate L4 for f(x)=x^2-x+1 and 1<=x<=3. Furthermore, on the given axes, sketch the graph of f(x) and the rectangles corresponding to L4.

Break the interval 1≤x≤3 into 4 subintervals of length Δx=0.5, starting at 1, 1.5, 2, and 2.5. Find f(x)=x²-x+1 at these points:

f(1)=1, f(1.5)=1.75, f(2)=3, f(2.5)=4.75.

Then the lower sum is

L

_{4}=(1+1.75+3+4.75)*0.5=5.25.Notice how much the 4 rectangles underestimate the area under the graph, especially the rightmost one.

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