I am currently working on the problem below and I am in need of help.
Consider the definite integral $\int_{1}^{2}\frac{1}{t}dt$.
(a)By the dividing the interval $1\leq t\leq 2$ into $n$ equal parts and choosing appropriate sample points, show that $$\sum_{j=1}^{n}\frac{1}{n+j}< \int_{1}^{2}\frac{1}{t}dt< \sum_{j=0}^{n-1}\frac{1}{n+j}$$ (b)How large should $n$ be to approximate $\int_{1}^{2}\frac{1}{t}dt$ with an error of at most $5(10^{-6}) $ using one of the sums in part (a)?
Hint: What is the difference between the underestimate and over estimate?