Find the surface area of the given surface. The portion of the cone z=√(x^2+y^2) below the plane

Find the surface area of the given surface. The portion of...

Key Concepts

Parametrization and Coordinate Transformation

Parametrization is the process of expressing a surface in terms of alternative variables, which can simplify calculations, especially for surfaces with symmetry. Coordinate transformations, such as switching to polar or cylindrical coordinates, are often employed to simplify the integration limits and the integrand by taking advantage of the inherent symmetry in the region of integration.

Bounds of Integration

Determining the correct integration limits is key to accurately calculating the surface area. The bounds are typically derived from the conditions or intersections provided by the surfaces in the problem. A clear understanding of how the surfaces intersect or are constrained ensures that the integration covers the appropriate region.

Surface Area of a Graph

This concept involves finding the area of a surface defined as the graph of a function z = f(x, y) over a region in the xy-plane. The area can be computed by integrating the differential area element, which is typically formulated as ?(1 + (?f/?x)² + (?f/?y)²) dx dy. This method is essential when dealing with surfaces described explicitly as a function of two variables.

Double Integrals

Double integrals are used to accumulate quantities over a two-dimensional region, and in the context of surface area, they allow the integration of the differential area element over the entire region of interest. Mastery of setting up and evaluating double integrals is crucial for computing areas and other properties of surfaces.

Transcript

00:01 In this video, we're given that z is the square root of x square plus y squared, and z goes from 0 to 4.

00:09 And we're asked to determine the surface area.

00:12 So if we're going to draw this, if we're going to graph it, we get an inverted cone.

00:18 All right, so the first thing we're going to do is we're going to attempt to parameterize this code.

00:23 So we can let x equal to r cosine theta and y equal r sine theta and z is equal to r.

00:32 And now this is the right parameterization, because as you see, the cone is just a circle that keeps expanding, and it goes from zero all the way up to four.

00:44 All right, so the second thing we're going to do is we're going to take the derivative of r with respect to theta, and then the derivative of r with respect to r.

00:52 So the derivative of r cosine theta with respect to theta is just negative r -sign theta.

00:57 The derivative of r -sign theta with respect to theta is just r -coside theta, and the derivative of r -cosine theta.

01:03 With respect to theta is zero because there are no theta in r.

01:06 And now the derivative of r with respect to r, so the derivative of r cosine theta with respect to r is just cosine theta, derivative of r sine theta with respect to r is just sine theta, the derivative of r is simply one.

01:20 Now we're going to determine the cross product.

01:24 So now we just arrange this neatly.

01:26 So in our first row we're going to put the r sub theta, right here, we're going to put this in our first row, second, sorry, the second row we're going to put r theta, and the third row we're going to put r sub r.

01:40 So we get something like this.

01:42 So i hat underneath that we're going to have negative r sine theta and then cosine theta, j hat underneath that we're going to have r cosine theta, then sine theta, k hat underneath that we're going to have zero and one.

01:53 All right, so let's determine the cross product.

01:55 So the first thing we do is we cancel this, we cancel this, and now we find it a determinant.

02:00 So r cosine theta times 1 minus sine theta times 0.

02:05 So it's just r cosine theta.

02:07 And then we tack on an i hat.

02:10 And now the j hat.

02:12 So we cancel this.

02:13 We cancel this.

02:14 Now we find a determinant...