approximating area, sigma sum/limit equation

Dec 2009
103
0
The problem
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?
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
That "1" is the initial x value. Using your \(\displaystyle x_n= \frac{15i}{n}\), when i= 0, \(\displaystyle x_0= 0\) and when i= 15, \(\displaystyle x_15= 15\). But you want the integral to go from 1 to 16, not 0 to 15.

In general, if the integral is from a to b, with n steps, you want the step to be \(\displaystyle \frac{b- a}{n}\) but then the x values will be \(\displaystyle a\), \(\displaystyle a+ \frac{b-a}{n}\), \(\displaystyle a+ \frac{2(b-a)}{n}\), \(\displaystyle \cdot\cdot\cdot\), \(\displaystyle a+ \frac{i(b-a)}{n}\), \(\displaystyle \cdot\cdot\cdot\).
 
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Dec 2009
103
0
what if y=c?
I have a problem where I have to find the integral from 2 to 3 and the points of the graphs are (2,3) and (3,3) so y=3 or f(x)=3
when evaluating the integral, i know that there is a rule where integral of a to b =c dx =c(b-a)
but if you didn't know that rule and just did it the long way with the reiman's sum method, would you still have to add that initial x? in this case 2?
I tried using the reiman's sum method and adding 2 and I got the wrong answer. 5 instead of 3.