Consider $I = \int^{14}_{8} e^{-x^4} dx$. Match each riemann sum.

Question

$I = \int^{14}_{8} e^{-x^4} dx$

We have $L_{1000}, L_{10}, R_{1000}$.

We have $I= 0.335, 0.368, 0.367$.

Match each sum with each $I$ value.

I know that the graph is decreasing and approaches $0$ on the interval $[8,14]$, so $R_n \leq \text{ Actual Area I} \leq L_n$.

I know $R_{1000} = 0.335$, as it is the number most away from the actual area $I$.

I think $L_{10} = 0.367$ and $L_{1000} = 0.368$ since $1000$ rectangles is much more accurate then $10$ rectangles, and thus it will be more larger then the actual area $I$.

Is this correct? It's the first time I'm doing these types of questions so I'm not $100%$ sure.

(not real approximations)

Answer 1

Since the function is decreasing.

$L_n \ge \int_a^b f(x) dx \ge R_n$

The right sum is less than the true value. The left sum is greater than the true value.

As $n$ gets larger $L_n$ gets closer to the true value.

Your logic is good there.

Except then $L_{n}\ge L_{n+1}$