Use Green's Theorem to find the area of each region. Bounded by the line x+y=3 and the hyperbola

Use Green's Theorem to find the area of each region. Bounded...

Key Concepts

Curve Intersection and Boundary Analysis

Analyzing the intersections of curves is vital to properly define the limits and segments of the region’s boundary. Understanding where and how the curves meet ensures that a closed, continuous boundary is established, which is necessary for correctly applying Green’s theorem and calculating area.

Parameterization of Curves

Parameterization involves expressing the coordinates of points on a curve in terms of a single variable, which is critical when evaluating line integrals. This technique allows the boundary of the region to be described in a form that makes the application of Green’s theorem computationally straightforward.

Green's Theorem

Green’s theorem is a central result in vector calculus that connects a line integral around a closed, simple curve to a double integral over the region it encloses. This theorem provides a powerful method for converting a complex area or circulation problem into a more manageable integral along the boundary of the region.

Line Integrals for Area Calculation

Utilizing line integrals to calculate areas involves defining the boundary of a region and then integrating a function along that boundary. By choosing specific functions compatible with Green’s theorem, the double integral representing the area can be transformed into a line integral, streamlining the area computation.

Transcript

00:01 For this problem, we are asked to find, or to use green's theorem to find the area of the region bounded by the line x plus y equals 3, and the hyperbola xy equals 2.

00:11 So, we are trying to find the area of this region enclosed, where the red line corresponds to x plus y equals 3, and the blue line corresponds to x, y, equals 2.

00:24 Which i'll note, we can rewrite both of these as y equals 3 minus x and y equals 2.

00:31 Over x respectively so what we'll do is now keep in mind we want to do this using greens theorem explicitly what we'll do is split this apart into two different curves i'll label them c1 and c2 and then use green's theorem to find the contribution to the area by integrating along each curve so we have c1 c2 so to begin with c1 we can define and we do want to be if uh if uh you for no reason other than consistency, let's go counterclockwise here.

01:07 That is the standard for greens theorem to make sure everything works out properly.

01:10 Let's say that we have x equals well it starts at 2.

01:14 So let's say x equals 2 minus t and it ends up at 1 so t is between 0 and 1.

01:21 And then we have y equals 3 minus x but x is 2 minus t so y equals 3 minus 2 minus negative t.

01:29 So 3 minus 2 plus t or y equals t plus 1, which would then give us that dx equals negative d t and d y equals d t.

01:40 So we have from green's theorem that the area will be equal to, in this case it's the integral from 0 to 1, or 1 half the integral from 0 to 1, of negative y d x.

01:53 So that would be negative t minus 1 times negative d t.

01:57 So just t plus 1...

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