Function Values

Question

 

                                                                                        6

For the function whose values are given in the table below,  ∫ f(x)dx  is approximated by a Riemann Sum using the

                                                                                       °

 

value at the midpoint of each of three intervals with width 2.

 

x     0     1       2     3      4     5   6
f(x) 0  0.25 0.48 0.68 0.84 0.95  1

The approximation is: (2 points)

1.40

1.88

3.65

3.76

Answer 1

3.76 is the answer.

 

To find the midpoint Riemann sum, you take the width of the subinterval, n, which is 2 in this case (it's also called delta x) and you multiply it by the sum of f(x) at each midpoint in the interval.

Since the domain is [0,6] and the subintervals have a width of 2, the midpoints are 1, 3, and 5 (in between 0 and 2 is 1, between 2 and 4 is 3, and between 4 and 6 is 5).

Therefore, at x = 1, f(x) = .25 (this value is from the table), at x = 3, f(x) = .68, and at x = 5, f(x) = .95. Add these values of f(x) together and multiply that value by 2 to obtain 3.76.