Use Definition 2 to find an expression for the area under the graph of f as a limit. do not evaluate the limit.

Definition 2 is

\(\displaystyle A = \lim_{n->\infty} R_{n}= \lim_{n->\infty} [f(x_{1})\Delta x+f(x_{2})\Delta x+....+f(x_{n})\Delta x]\)

17. \(\displaystyle f(x)=\sqrt[4]{x}, 1 \leq x \leq 16 \)

I got

\(\displaystyle \lim_{n->\infty} \sum^n_{i=1} [f(x_{i}) \times

\frac{b-a}{n}]

\)

\(\displaystyle =\lim_{n->\infty} \sum^n_{i=1} [f(x_{i} \times \frac{15}{n}]\)

\(\displaystyle =\lim_{n->\infty} \sum^n_{i=1} [f(\frac{15i}{n})\times\frac{15}{n}]\)

\(\displaystyle =\lim_{n->\infty} \sum^n_{i=1} [\sqrt[4]{\frac{15i}{n}}\times \frac{15}{n}]\)

but the book's answer is

\(\displaystyle \lim_{n->\infty} \sum^n_{i=1} [\sqrt[4]{1 + \frac{15i}{n}}\times \frac{15}{n}]\)

where did that extra 1 come from?