Thread: Riemann Sum to an integral

1. Riemann Sum to an integral

I'm having problem with this problem. I've attached a picture, because I cannot figure out how to simplify this down to typing.

I've tried just about everything our teacher has taught/shown us, but I'm coming up with wrong answers.

2. Remember the general form of this:

$\displaystyle \lim_{n->\infty}\Sigma_{k=1}^nf(a+k\Delta x)\Delta x =\int_a^bf(x)dx$

$\displaystyle \Delta x=\frac{b-a}{n}$

In this case, the $\displaystyle \Delta x=\frac{3}{n}$ since $\displaystyle a=0$. In the case that $\displaystyle f(x)=sin(x)$ we have:

The $\displaystyle sin(a+k\Delta x)=sin(k \Delta x)=sin\left(k\frac{3}{n}\right)$

However, you're given $\displaystyle sin\left(k\frac{9}{n}\right)$, so what do you have to do to $\displaystyle sin(x)$ to get $\displaystyle sin\left(k\frac{9}{n}\right)$ instead of $\displaystyle sin\left(k\frac{3}{n}\right)$?

3. You could multiply by 3? I just got the answer right after you replied. Thanks though. The way I did it, it didn't really stick in my head. But after reading this I feel like I've learned something. Thanks.