how do you draw paraboloid using z=1-x^2-y^2
sorry I haven't taken math for 2 years. I forget how to. I would really appreciated if you can explain me
this is the whole problem
use cylindrical coordinates. Evaluate ∫∫∫E (x^3 +xy^2)dV, where E is the solid in the first octant that lies beneath the paraboloid z= 1-X^2-y^2
I am trying to solve this problem but i feel like I won't know the bound until i know how to draw the paraboloid
To visualize the parabaloid, rewrite as , and you can see that at a "height" z above the origin, we have a circle of radius . So we have a unit circle at coming to a point at to form a 3-D bell shape. To use just the first octant, we will quarter the bell much like you would quarter an apple, cutting it into four equal segments from the top. One of these segments represents the solid E.
Now convert to cylindrical coordinates:
To represent the region in cylindrical coordinates, start by visualizing ranging from to on our quartered bell, ranging from the "center" to the edge . Finally, our inequality takes the form , so a point at a distance from the z-axis can be at any height between . Therefore, . These are your bounds of integration.
Can you take it from there?