How to find the area of a triangular region using Green's Theorem

[Mohamed Abdul]

Homework Statement

You have inherited a tract of land whose boundary is described as follows. ”From the oak tree in front of the house, go 1000 yards NE, then 1200 yards NW, then 800 yards S, and then back to the oak tree.

Homework Equations

Line integral of Pdx + Qdy = Double integral of partialQ/partialx - partialP/partialy

The Attempt at a Solution

I'm very confused as how to approach this function, given that I don't have any equations to work with. Do I start by having the first thousand yards extend from the origin, and then find the coordinates of the lines? I don't really know.

 

[BvU]

Hi,

I'm confused with this exercise too: there is nothing that is being asked. What's the problem in the problem statement ?

 

[Ray Vickson]

Homework Statement

You have inherited a tract of land whose boundary is described as follows. ”From the oak tree in front of the house, go 1000 yards NE, then 1200 yards NW, then 800 yards S, and then back to the oak tree.

Homework Equations

Line integral of Pdx + Qdy = Double integral of partialQ/partialx - partialP/partialy

The Attempt at a Solution

I'm very confused as how to approach this function, given that I don't have any equations to work with. Do I start by having the first thousand yards extend from the origin, and then find the coordinates of the lines? I don't really know.

Bad title: the region is not triangular. (Draw it and see for yourself.)

 

[Dr.D]

Green's Th is very useful for calculating irregular areas. You simply have to make appropriate choices for the functions P and Q. I suggest that you start by imposing a rectangular Cartesian coordinate system on the whole problem and then set up the integrals required for the area.

 

[Mohamed Abdul]

Hi,

I'm confused with this exercise too: there is nothing that is being asked. What's the problem in the problem statement ?

It just states to use Greene's Theorem to find the area. The thing is usually with Greene's Theorem we're given functions so I'm not sure how to proceed.

 

[BvU]

It just states to use Greene's Theorem to find the area. The thing is usually with Greene's Theorem we're given functions so I'm not sure how to proceed.

What functions could be useful in this context (as @DrDu hints at already) ?

 

[Dr.D]

I assume that you know how to express the area as a double integral. Write that out. That gives you the form for the derivatives of P and Q (hint: it is often simpler to choose one of P or Q as simply zero). The work back to what P and Q must be for the boundary integral.

 

[Mohamed Abdul]

I assume that you know how to express the area as a double integral. Write that out. That gives you the form for the derivatives of P and Q (hint: it is often simpler to choose one of P or Q as simply zero). The work back to what P and Q must be for the boundary integral.

I'm having trouble constructing the integral given that I have four different lines on my cartesian plane. Usually I just have to deal with 2 separate functions, but here it looks like there are four.

 

[Dr.D]

A triangle should only involve three lines (three angles and three sides). Please look again.

 

[Ray Vickson]

I'm having trouble constructing the integral given that I have four different lines on my cartesian plane. Usually I just have to deal with 2 separate functions, but here it looks like there are four.

So, the calculation will require twice as much work as you have done before, but the principles are no different from what you have done already.

 

[Mohamed Abdul]

A triangle should only involve three lines (three angles and three sides). Please look again.

This is the image I got, I'm not sure how to proceed from here though.

 

[Ray Vickson]

View attachment 214427
This is the image I got, I'm not sure how to proceed from here though.

What are the $P$ and $Q$ that you want to use? What is preventing you from integrating $P dx + Q dy$ over each of the four line segments? You MUST TRY: we cannot do your work for you.