# Finding the value of the right-endpoint Riemann sum.

Calculate the area between $$𝑓(𝑥)=𝑥^2$$ and the x axis over the interval [3,12] using a limit of right-endpoint Riemann sums:
Find the value of the right-endpoint Riemann sum in terms of n:
$$\sum_{k=1}^n f(x_k)\Delta x=$$
I got $$81+\frac{243(n-1)}{n}+\frac {729(n-1)(2n-1)}{(6n^2)}$$ but it comes up as wrong.

• I got the same as you except for $+$ where you had $-$ Jan 24, 2020 at 0:39

Let $$x_k=3+\dfrac{9k}n$$ and $$\Delta x=\dfrac9n$$, so
$$\sum_{k=1}^n f(x_k)\Delta x=\sum_{k=1}^n \left(3+\dfrac{9k}n\right)^2\dfrac9n=\sum_{k=1}^n \left(9+\dfrac{54k}n+\dfrac{81k^2}{n^2}\right)\dfrac9n$$
$$=\left(9n+\dfrac{54}n\dfrac{n(n+1)}2+\dfrac{81}{n^2}\dfrac{n(n+1)(2n+1)}6\right)\dfrac9n,\\$$
which approaches $$(9+27+27)9=567$$ as $$n$$ approaches $$\infty$$.
Your answer also approaches $$567$$ as $$n\to\infty,$$ but you had $$-$$ where I have $$+$$.