Double Integral Limits for a Sideways Rectangle

In summary, the conversation discusses setting up limits for solving an integral within a tilted rectangle using Green's theorem. The suggested method involves using parallel lines through the vertices of the rectangle and changing variables to u and v. The importance of calculating the Jacobian is also mentioned.
  • #1
Moragor
7
0

Homework Statement


I have to solve the integral of (ytan^-1(x) - 3) inside the area of the rectangle with vertices (1,0), (0,1), (2,3), (3,2). How do I set up these limits?


Homework Equations


This is a tilted rectangle so I can't use just values for the limits?


The Attempt at a Solution


This is a Green's theorem problem that started out as integral(tan^-1(x)dx + 3xdy). I just can't figure out the limits.
 
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  • #2
The parallel lines through (1,0), (3,2) and (0,1),(2,3) are y= x-1 and y= x+ 1, respectively; both are variations on "y-x= constant". The parallel lines through (1,0),(0,1) and (3,2),(2,3) are y= 1-x and y= 5-x, respectively; both are of the form "y+ x= constant. Changing variable to u= y- x and v= y+ x gives an integral with limits of integration in u of -1 to 1 and in v, 1 to 5. Be sure to calculate the Jacobian to convert dxdy.

Have you already calculated the integration around the boundary?
 

Related to Double Integral Limits for a Sideways Rectangle

What is a double integral?

A double integral is a type of mathematical operation that involves finding the volume under a surface in three-dimensional space. It is essentially the combination of two single integrals, where the limits of integration are given by a region in the xy-plane.

What is a sideways rectangle?

A sideways rectangle is a type of two-dimensional shape that has a longer length than width, and is oriented horizontally. It can also be referred to as a horizontal rectangle or a rectangle in the plane.

How do you determine the limits for a sideways rectangle in a double integral?

The limits for a sideways rectangle in a double integral are determined by the region in the xy-plane that the rectangle covers. The lower limit will be the left boundary of the rectangle, and the upper limit will be the right boundary. The inner integral will have the lower and upper limits as the bottom and top boundaries, respectively.

What are the advantages of using double integral limits for a sideways rectangle?

Using double integral limits for a sideways rectangle allows for more flexibility in calculating the volume under a surface. It also allows for the integration of more complex shapes, such as curved surfaces, that cannot be easily calculated using single integrals.

Can double integral limits for a sideways rectangle be applied to real-world problems?

Yes, double integral limits for a sideways rectangle can be applied to real-world problems in various fields such as physics, engineering, and economics. For example, it can be used to calculate the volume of a three-dimensional object or to determine the amount of fluid flowing through a curved pipe.

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