1. The graph of the function f(x) on [0, 1] is given below:
Using this graph:
b. Determine how large an error there might be between your estimate and the actual value of this integral. In addition, explain why this error is positive or negative. That is, does your estimate overestimate or underestimate the value of the integral?
Solution: (a)
To estimate the integral form the left Riemann sum, the rightRiemann sum, and then take the average of the two as the estimate.
You will need to estimate, visually, the functional values from the graph. One such set of values is
|
x |
0 |
0.2 |
0.4 |
0.6 |
0.8 |
1 |
|
f(x) |
0 |
0.24 |
0.48 |
0.68 |
0.82 |
1.0 |
(Each of the f(x) values has was recorded with an error of, at most 0.02, and these recording errors are ignored in what follows)
Then:
Left_riemann = (0 + 0.24 + 0.48 + 0.68 + 0.82)(0.2) = 2.22*0.2 = 0.444
Right_riemann =(0.24 + 0.48 + 0.68 + 0.82 + 1)(0.22) = 3.22*0.2 = 0.644
Thus, the value of the integral is, approximately, 0.544.
b.
Since the left riemann sum L underestimates the integral and since the right,R, overestimates it we have
Finally, since the average of the left and right riemann sums is equal
to the trapezoidal approximating sum, and since f(x) is concave down, the average will underestimate the integral.