Identifying left- and right-Riemann sums of $\int_9^{14}e^{-x^4}\ dx$ 
My attempt: 

Relooking at it, I think $L_{20}$ would be the highest, so like
$R_{1200} < L_{1200} < L_{20}$, but I have no way to justify it, any help is appreciated. 

The main ideas here are that, for a decreasing function,


*

*Estimating using left endpoints (i.e., $L_{20}, L_{1200}$) will always overestimate the true value of the integral. 

*Estimating using right endpoints will always underestimate the value of the integral.
These two tell you that $R_m < I < L_n$ for any integers $m, n$.


*

*Using more subintervals (i.e., larger $n$) will always produces a better estimate -- they'll produce estimates closer to the actual value of the integral.


So, if $m < n$, we have
$$R_m < \underbrace{R_n < I < L_n}_{\text{more subintervals}} < L_m$$
which agrees exactly with what you have! (in the body of the question, not the image)
