Urgent! Calculus Help!

Question

1. 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.

Answer 1

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

L₄=(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.