Question

We have already derived the integral formulae for the mass m, the moment My about the y-axis, and the moment Mx about the x-axis, of the region R where a lamina with density ρ(x) resides in the xy-plane. The method we used was to:

-Slice R into n rectangles, where y = f(x) bounded R above and y = g(x) bounded R below, on [a, b].

-Compute the area, mass, and moments (about both the y-axis and the x-axis), of the i th rectangle Ri .

-Take the Riemann sum limit to derive the integral formulae for m, My, and Mx.

There are analogous integral formulae for m, My, and Mx, of R in terms of y (in class we did it in terms of x). Indeed now assume the region R is bounded to the right by x = f(y) and to the left by x = g(y) on [c, d] with density ρ(y).

Adapt the method we did in class to derive the formulae for m, My, and Mx, as y-integrals.

You must label or define relevant variables and quantities, and at the end take the Riemann sum limit.

Note: Only by replacing x with y in the x-integral formulae does not yield the correct y-integral formulae.

please please focus on "note" and it is also for "y integral"

I posted the question earlier but the answer was not the professor is looking for

Answer 1