# Example 5 approximate the area shown in the margin

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Example 5 . Approximate the area shown in the margin between the graph of f and the x -axis spanning the interval [ 2, 5 ] by summing the areas of the rectangles shown in the lower margin figure.
306 the integral lengths Δ x k for k = 1 , 2 , 3 , . . . , n . The mesh or norm of a partition P is the length of the longest of the subintervals [ x k - 1 , x k ] or, equivalently, the maximum value of Δ x k for k = 1, 2, 3, . . . , n . For example, the set P = { 2, 3, 4.6, 5.1, 6 } is a partition of the interval [ 2, 6 ] (see margin) that divides the interval [ 2, 6 ] into four subintervals with lengths Δ x 1 = 1 , Δ x 2 = 1.6 , Δ x 3 = 0.5 and Δ x 4 = 0.9 , so the mesh of this partition is 1.6 , the maximum of the lengths of the subintervals. (If the mesh of a partition is “small,” then the length of each one of the subintervals is the same or smaller.) Practice 6 . P = { 3, 3.8, 4.8, 5.3, 6.5, 7, 8 } is a partition of what inter- val? How many subintervals does it create? What is the mesh of the partition? What are the values of x 2 and Δ x 2 ?
Example 6 . Find the Riemann sum for f ( x ) = 1 x using the partition { 1, 4, 5 } and the values c 1 = 2 and c 2 = 5 (see margin).
Practice 7 . Calculate the Riemann sum for f ( x ) = 1 x on the partition { 1, 4, 5 } using the chosen values c 1 = 3 and c 2 = 4.
4 . 1 sigma notation and riemann sums 307 Practice 8 . What is the smallest value a Riemann sum for f ( x ) = 1 x can have using the partition { 1, 4, 5 } ? (You will need to choose values