00:01
So in this question, we're going to use a double integral in a computer algebra system to find the volume of the solid.
00:08
The solid is bounded above by the paraboloid, z equals 1 minus x squared minus y squared, and below by the xy plane.
00:17
So i have a paraboloid opening downwards.
00:24
Its vertex is it 0 .01 opening downwards, and it's hitting the xy plane.
00:33
And i'm trying to find the volume that lies beneath the paraboloid and above the xy plane.
00:43
So to find my volume, i'm going to say that i'm going to compute the double integral over r of 1 minus x squared, minus y squared d a where my r is the region in the xy plane that is beneath this paraboloid so i need to figure out what the equation of that bounding circle is well the paraboloid is going to hit the xy plane remember the xy plane is where z equals zero when zero equals one minus x squared minus y squared when x squared plus y squared equals one so my region r down in the x y plane is going to look something like this and so let's now set up limits of integration for this i'm going to have the double integral we set of one minus xx squared minus y squared, and i'll set this up, for sake of argument, in the dydx direction.
02:09
Okay? so in the dydx direction, i look at a line parallel to the y axis.
02:18
Where would such a line enter my region? well, if i have x squared plus y squared equals 1, i have y squared equals 1 minus x squared, y equals plus or minus the square root of 1 minus x squared.
02:40
So my negative square root is the bottom half of this circle, while the positive square root is the top half of this circle.
02:57
So my y's here, they are ranging from negative the square root of 1 minus x squared.
03:05
That's where i'm entering my region to positive the squared of 1 minus x squared.
03:13
That's where i leave my region.
03:18
And i am doing this from x equals negative 1 to 1...
