Question

How to determine which of the following regions has an area equal to the given limit without evaluating the limit ?

Answer 1

This is the limit of a Riemann Sum for the function `f(x)=x^10` on the interval `[5,7]`

Explanation:

A Riemann sum has the form

`sum_(i=1)^n f(x_i)Deltax`

Where `Deltax = (b-a)/n`. We can interpret this expression as having `Deltax = 2/n`, so `b-a=2`.

If we partition an interval `[a,b]` into `n` subintervals of equal length, then we can find the left endpoints of the subintervals using `a+iDeltax` to find the right endpoint of subinterval `i`.
In this expression we can interpret `5+(2i)/n` as this calculation.

Finally, we see the the expression has `(x_i)^10`, so we must have `f(x)=x^10`

Therefore, it represents the area under the curve `y=x^10` and above the `x`-axis between `x=5` and `x=7`.