Calculus Riemann sums for circle and ellipse

Question

I ran into this problem today. I need to compare the Riemann sums for a circle and an ellipse. I have no idea as where to start.

Here's the question:

Answer 1

Hint: write out the Riemann sum for the area of the circle. One of the factors in each summand in that sum is the $y$-coordinate of the circle. The exercise tells you how to write the corresponding Riemann sum for the ellipse. Do so, then see if you can simplify the resulting sum to be a constant times the sum representing the area of the circle.

So, the equation of the upper half-circle is $f(x)= y = \sqrt{a^2-x^2}$. Let $g(x)$ be the equation of the upper half of the ellipse; from the exercise statement, $g(x) = \frac{b}{a}f(x) = \frac{b}{a}\sqrt{a^2-x^2}$. Then the two Riemann sums for the area of the upper halves are $$\sum_{i=1}^N x_i f(x_i)\Delta x,\qquad \sum_{i=1}^N x_ig(x_i)\Delta x.$$ If you write those out explicitly and look at the one for the ellipse, you should be able to take it from there.