1. ## Two Questions

1. When asked to sketch the region of $\displaystyle {(x,y)|x^{2}\le y\le 2}$
I'm pretty sure this is just an oval with it's two halves being $\displaystyle x^2 from -\sqrt{2} to \sqrt{2}$ but I'm not sure if thats right.
2. I'm suppose to set up a proper and improper integral to find the length of a curve $\displaystyle y^3 = x from (-1,-1) to (8,2)$.
I believe I've found the proper one to be $\displaystyle \int_{-1}^{2} \sqrt{1+9y^4} dy$
Any hints on what the improper one might be?
Thanks in advance for the help.

2. It's not real obvious that you have or have not the first one correctly determined. The way I might do it...

Graph y = 2

Graph x^2 = y

This divides the x-y plane into 5 distinct sections. Just pick a poitn from all 5 and see if it works.

Above the line and inside the parabola

(0,3)

Above the line and right of the parabola

(3,3)

Above the line and left of the parabola

(-3,3)

Below the line and inside the parabola

(0,1)

Below the line and outside the parabola

(2,1)

Let's see

(0,3): o^2 = 0 <= 3 <= 2 -- Nope.
(3,3): 3^2 = 9 <= 3 -- Nope.
(-3,3): (-3)^2 = 9 <= 3 -- Nope.
(0,1): 0^2 = 0 <= 1 <= 2 -- Yes!
(2,1): 2^2 = 4 <= 1 -- Nope.

That settles it. Below the line and inside the parabola.