Use a parametrization to express the area of the surface as...

Use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the text. They should have the same values, however.) The portion of the plane $y+2 z=2$ inside the cylinder $x^{2}+y^{2}=1$

Use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the text. They should have the same values, however.)
The portion of the plane $y+2 z=2$ inside the cylinder $x^{2}+y^{2}=1$

Key Concepts

Projection of Surfaces onto Coordinate Planes

In many surface area problems, it is beneficial to project the surface onto a simpler plane (like the xy-plane) where the integration region is easier to define. The differential area element is then modified by a factor that accounts for the orientation of the surface relative to the projection plane. This technique is a cornerstone in the computation of surface integrals in multivariable calculus.

Parametrization of Surfaces

This concept involves expressing a surface in terms of two independent parameters, which makes the computation of properties such as area more manageable. In the broader subject of multivariable calculus, parametrization is a key tool for converting complex surfaces into a form that is easier to integrate over, by representing each point on the surface as a function of two parameters.

Surface Area Integrals

Surface area integrals are used to compute the area of surfaces embedded in higher-dimensional spaces. The standard method involves either using the cross product of the partial derivatives of the parametrization to obtain the differential area element or applying formulas for the surface area of graphs. This concept is essential in calculus for finding measures of curved surfaces.

Change of Variables and Coordinate Systems

When setting up integrals for regions bounded by curves or surfaces, converting to alternative coordinate systems, such as polar coordinates, can simplify the computation. This involves a change of variables where the integration domain is re-expressed in a coordinate system that better matches the geometry of the problem, and it requires the use of a Jacobian determinant to correctly adjust the area element.

Transcript

00:01 So you have up the following surface, well is a portion inside of this cylinder.

00:15 X square plus y square equals one, that is a cylinder in the x.

00:19 In three -dimensional space, it's a cylinder.

00:24 A unit cylinder that has a unit circle has the face and it goes all along, it extends up.

00:36 Along c, or this is under the convention that we have to the x -axis, the c in the y.

00:48 It is inside of this, in that and along this plane, along the plane given by y plus 2c is equal to 2.

01:03 So this is a plane that intersects here.

01:10 So it's equal to 0, y should be 2.

01:15 It contains this point and then y is equal to 0c should be 1.

01:22 So it is a plane going through this point and then and extends parallel to the x -axis.

01:36 So we need the cut, it should be a region like this, like on the ribs.

01:50 So this is the region.

01:52 The green region are we want to find the area.

01:58 The area.

01:59 So for that we need to make a parameterization of the surface.

02:04 Of the surface so let's call it uh...

02:09 You and v these new arrivals and so well let's just say that u s equal to x and v is equal to y so that um.....

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