Right Riemann Sum limit question URGENT

Question

I have no idea how to find the last question regarding the limit of these Reimann sums as n approaches infinity. Please help! Here is the link to the example https://ibb.co/TbGg4Hs

Answer 1

Riemann sums approximate the area under a curve. For a continuous function, Right-hand Riemann sums with n subintervals of equal length, (which is what you have here - n subintervals of [0,4] of length 4/n), always converge to the true area under the curve - the integral of the function on the same interval.

It is possible to find an anti-derivate for sqrt(16-x^2), but for this problem, once you know the limit of the Riemann sums is the area under the curve (the definite integral), you can just use the fact that you are looking for the area of the top-right quarter of a circle of radius 4.

So, you can find the area of a circle of radius 4 and take a quarter of that.