help on this problem?

I'd do 2 Riemann sums (left-hand and right-hand) and take the average.

(3 * (1.26 + 1.99 + 5.47 + 1.59 + 1.71 + 2.69) + 3 * (1.99 + 5.47 + 1.59 + 1.71 + 2.69 + 3.76)) / 2 =>

3 * (1.26 + 2 * 1.99 + 2 * 5.47 + 2 * 1.59 + 2 * 1.71 + 2 * 2.69 + 3.76) / 2 =>

3 * (1.26 + 3.76 + 2 * (1.99 + 5.47 + 1.59 + 1.71 + 2.69)) / 2

3 * (5.02 + 2 * (7.46 + 3.3 + 2.69)) / 2

3 * 2 * (2.51 + 7.46 + 5.99) / 2

3 * (9.97 + 5.99)

3 * 15.96

3 * (16 - 0.04)

48 - 0.12

47.88

Different integration rules give different results.

One of the easiest rules to use is trapezoidal integration. Add all of the y-values except the first and last. That sum is 13.45.

Now, double this value and add the first and last y-values. The y-value total is 31.92.

Multiply this by (21-3)/12 = 3/2 to get the value of the integral, 47.88.

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If you use Simpson's rule, you will add twice the sum of the 2nd, 4th, and 6th y-values to the above y-value total. This gives you 44.46. The final integral is this value multiplied by (21-3)/18 = 1 to get 44.46.

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Simpson's 2nd rule adds the 2nd, 3rd, 5th, and 6th y-values to the trapezoidal rule y-value total. That sum is 43.78. The final integral is this value multiplied by (21-3)/16 = 9/8 to get 49.2525.

The graphic at the source link shows the graph that each of these methods assumes. The lower right graph is a spline interpolation that is more complicated than I want to describe here. It is shown for your edification.