# Rewriting this Riemann sum as a definite integral

Can someone help me rewrite this Riemann sum as a definite integral?

$$\displaystyle \lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^{n}\left(-7+\frac{14i}{n} \right)^9\sin\left(4+ \left(-7+\frac{14i}{n} \right)^8 \right)$$

$$\Delta x=\frac{1}{n}$$ so this means $$b-a=1$$. If I rewrite the $$\frac{14i}{n}$$ as $$14\frac{i}{n}$$, then this means I could say that $$a=0$$ and $$b=1$$ so this means my definite integral is:

$$\int_{0}^{1}(-7+14x)^9\sin(4+(-7+14x)^8)dx$$ but I am not sure if this is correct or not.

If this IS correct, how would I do this integral with no integration technique other than u-substitution or manipulation? I can't seem to do it otherwise.

Assuming partition points are of the form $$x_i=a+i \frac{b-a}{n}$$, it looks like $$a=-7$$ and $$b-a=14$$ so $$b=7$$. The $$1/n$$ that's pulled out of the sum is misleading. Rewrite it as $$1/n=\frac{1}{14}\cdot \frac{14}{n}$$. This leads to the integral $$\frac{1}{14}\int_{-7}^7 x^9 \sin(4+x^8)$$ Hint: To evaluate the integral, take note of the bounds and types of functions.