The purpose of this tutorial assignment is to give a clearer picture of the idea behind the Riemann integral as a measure of the area under a curve. Later in the course, we will see the Fundamental Theorem of Calculus, which gives a powerful method for finding these areas, but first we will examine the underlying idea.
Step 1: Make a large, careful graph of y = x2 - x + 1 on the interval (1,9).
Step 2: Divide the interval (1,9) into n = 4 equal subdivisions, each subinterval having width , with subdivision points x=3, x=7, and x=9. Make a table showing the intervals, their left endpoints midpoints , and right endpoints , and the values of at each of these points
Step 3: For each subinterval, draw rectangles with base on the x-axis and height ,, and respectively.
Step 4: Another way we could get a fairly good approximation to the area under the curve would be to take the average of and , that is, for the height of our rectangles. This is equivalent to taking trapezoids whose tops runs from
to .
Exercise 1: Repeat the above procedure for on the interval (1,9), using n = 8 subintervals, with ∆x = (9 - 1)/8 = 1.
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Calculate , and . What do you observe?