$$ \int_{1}^{9}(x^2+7) dx $$
That's the integral. And I'm supposed to evaluate it by using:
$$ \lim_ {x \to \infty} \sum_{i=1}^{n} f(x_i)(\Delta x) $$
Also $$ \Delta x = \frac{b-a}{n} $$
And $$ x_i = a + \frac{i(b-a)}{n} $$
The question asks: "Now find the sum of the areas of
n approximating rectangles. (Your answer will include variable n)"
My solution:
$$ \lim_ {n \to \infty} \sum_{i=1}^{n} \left[ \left(\frac{8i}{n}\right)^2 + 7\left(\frac{8i}{n}\right) \right] \cdot \frac{8}{n} $$
$$\lim_ {n \to \infty} \left(\left(\frac{8}{n}\right)^2 \cdot \frac{8}{n}\right) \sum_{i=1}^{n} [i^2] + \left(\left(\frac{8}{n}\right) \cdot \frac{8}{n}\right)\sum_{i=1}^{n} [ i ] $$
$$ \lim_ {n \to \infty} \left(\frac{8}{n}\right)^3 \left(\frac{n(n+1)}{2}\right)^2 + \left(\frac{8}{n}\right)^2\left(\frac{n(n+1)}{2}\right) $$
I've tried a few variations of this but nothing seems to work… The final answer is supposed to be 896/3 and the question I cited earlier calls for a simplified version of what I have up there, but this just isn't making sense at all.
Best Answer
The formula of the sum of first $n$ squares is mistakenly replaced by that of the sum of first $n$ cubes.
Using the elementary definition of the definite integral, $$ \int_a^b f(x) d x=\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{b-a}{n} f\left(a+\frac{k(b-a)}{n}\right) $$ Putting $a=1$ and $b=9$ and $f(x)=x^2+7$ yields $$ \begin{aligned} \int_1^9\left(x^2+7\right) d x & =\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{8}{n}\left[\left(1+\frac{8 k}{n}\right)^2+7\right] \\ & =64 \lim _{n \rightarrow \infty} \sum_{k=1}^n\left(\frac{1}{n}+\frac{2 k}{n^2}+\frac{8 k^2}{n^3}\right) \end{aligned} $$ Using the formula for sum of first $n$ positive integers and their squares, $$ \sum_{k=1}^n k=\frac{n}{2}(n+1) \text { and } \sum_{k=1}^n k^2=\frac{n}{6}(n+1)(2 n+1), $$ we get $$ \begin{aligned} \int_1^9\left(x^2+7\right) d x & =64 \lim _{n \rightarrow \infty} \sum_{k=1}^n\left(\frac{1}{n}+\frac{2 k}{n^2}+\frac{8 k^2}{n^3}\right) \\ & =64 \lim _{n \rightarrow \infty}\left(1+\frac{2}{n^2} \cdot \frac{n(n+1)}{2}+\frac{8}{n^3}\cdot \frac{n}{6}(n+1)(2n+1)\right) \\ & =64\left(2+\frac{8}{3}\right) \\ & =\frac{896}{3} \end{aligned} $$