Question 4.7.3: Finding an Approximation with a Given Accuracy Use the Midp...

To obtain the desired accuracy, we continue increasing n until it appears unlikely the third decimal will change further. (The size of n will vary substantially from integral to integral.) You should confirm the numbers in the accompanying table.

From the table, we can make the reasonable approximation

\int_0^2 \sqrt{x^2+1} d x \approx 2.958.

While this is reasonable, note that there is no guarantee that the digits shown are correct.

To get a guarantee, we will need the error bounds derived later in this section.