Sketch the region enclosed between the surfaces and describe their curve of intersection. The el

Sketch the region enclosed between the surfaces and describe...

Key Concepts

Intersection of Surfaces

The intersection of surfaces concept involves finding the set of points that satisfy the equations of two or more surfaces simultaneously. This often results in curves or other lower-dimensional shapes, and analyzing these intersections is crucial in multivariable calculus for understanding how different geometric objects interact in three-dimensional space.

Paraboloid

A paraboloid is another type of quadric surface commonly given by a quadratic equation in two variables equated to a linear term in the third. In its elliptic form, it appears as a bowl-shaped surface that opens along one axis, and is relevant in applications ranging from optics to computational geometry, particularly when analyzing surfaces formed by quadratic relationships.

Coordinate Transformation

Coordinate transformation, such as switching to polar or cylindrical coordinates, is a powerful tool when dealing with radially symmetric surfaces. This technique simplifies complex equations by reducing the number of variables or by taking advantage of the symmetry, making it easier to determine intersections and regions enclosed by multiple surfaces.

Quadric Surfaces

Quadric surfaces are the set of surfaces represented by second-degree polynomial equations in three variables. They include shapes such as ellipsoids, hyperboloids, paraboloids, and cylinders, which are studied in multivariable calculus and analytic geometry for their diverse geometric properties and applications in modeling physical phenomena.

Ellipsoid

An ellipsoid is a type of closed quadric surface that generalizes the sphere by allowing different scaling along each coordinate axis. Its equation, which is quadratic in x, y, and z, describes a symmetric, oval shape that is pivotal in understanding volume, surface area, and the effects of different scaling factors in geometric analysis.

Transcript

00:01 In this problem, we're given two quadratic surfaces, and we want to sketch the region enclosed by them and describe the curve of intersection.

00:13 Our surfaces are 2x squared plus 2y squared plus z squared equals 3, and z equals x squared plus y squared.

00:26 And so we want to describe these.

00:28 So we're going to need to recall some of the properties for ellipssoids and paraboloids.

00:36 So here we have an ellipsoid, here we have a probloid.

00:39 And we're also going to need to recall how we find a curve of intersection.

00:45 So first we're going to try to find the curve of intersection because it might make grapping easier.

00:49 So if we take our second equation and substitute it into our first, because we can rewrite our first equation.

00:59 So let's do here's one, here's two.

01:01 So by one, we have two x squared plus y squared plus z squared is equal to three.

01:09 So that's two z plus z squared minus three is equal to zero.

01:18 So let's reverse boil this.

01:19 So we get z plus three times z minus one is equal to zero.

01:25 So we get z is equal to negative three or one.

01:28 So it's kind of weird because we've got two solutions.

01:32 So let's look at this z equals 3.

01:35 And let's also look at the curve, z equals x squared plus y squared.

01:40 And since x and y are real numbers, negative 3 equals x squared plus y squared is a contradiction.

01:54 Therefore, because we can't have a negative value is equal to sum of two positive values.

02:00 It doesn't make any sense.

02:02 So therefore, we're going to throw out, we're just going to go ahead and throw out this three value.

02:08 We don't need that.

02:09 So we have z.

02:10 We're looking at the plane z equals 1.

02:13 And so either we can plug it into our first equation, but the second equation looks pretty easy.

02:18 Since we just have 1 equals x squared plus y squared.

02:22 So that's a circle with radius equal to 1.

02:28 And the center is going to be at the point 0 ,0, and what was the z? z is equal to 1 ?001.

02:39 Okay, so what is this shape going to look like? so we have this paraboloid that opens from the origin up, and then here we have this ellipsoid that kind of lives somewhere in this paraboloid.

02:59 And the only place where they intersect is going to be at a circuit.

03:07 Circle, so we're just trying to get a rough idea of this sketch...

Please give Ace some feedback