First, solve for y:
Since we are told to find the area under the ellipse in the first quadrant, we take
Now, we are to use a midpoint Riemann Sum:
We are told that , and the ellipse's major axis is along the x-axis and has a length of 4 units. Thus, we are applying the Riemann sum over the interval .
What is the midpoint of each section?
Our midpoint x values are
So our Riemann sum can be expressed as . Evaluating, we see that we have:
Thus, the total area would be 4 times this amount.
Does this make sense?