Integration over a region/(double integral, how to factor it) Urgent please

In summary, the conversation discusses using a double integral to find the area of a region in R2 described by a formula. The hint to complete the square is provided, but the individual is having difficulty completing the square and understanding the resulting form. They mention the form being a rotated ellipse and provide a link to a visual representation. They also mention not being able to use calculators and needing to factor by hand.
  • #1
seto6
251
0

Homework Statement


Use a double integral to find the area of the region D, where

D={(x,y) contained in R2 | 4x2+12xy+13y2+40y<=-75}

Hint: complete the square
I have a hard time getting the region, i should complete the square to get a formula in R2 that describes the region, i tried to do complete the square but it comes something like this:
(2x+3y)2 + 4(y+5)2=25

this does not make sense it look weird, i am not sure how to get this area. please help

The Attempt at a Solution


above
 
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  • #3
yes i have tried that but, I am not suppose to use any calculators. just by hand there fore some how i must factor not sure how
 

FAQ: Integration over a region/(double integral, how to factor it) Urgent please

What is integration over a region and why is it important?

Integration over a region, also known as a double integral, is the process of finding the volume under a surface in a three-dimensional space. It is important because it allows us to calculate the total amount or value of a function over a specific region, which has many practical applications in math and science.

What is the difference between a single integral and a double integral?

A single integral calculates the area under a curve in a two-dimensional space, while a double integral calculates the volume under a surface in a three-dimensional space. In other words, a single integral deals with two-dimensional objects, while a double integral deals with three-dimensional objects.

How do you set up a double integral?

To set up a double integral, you first need to identify the limits of integration, which define the boundaries of the region you are integrating over. Then, you need to determine the order of integration, which dictates the order in which you will integrate with respect to the variables in the function. Finally, you need to write the integrand, which is the function you are integrating, in terms of the chosen variables.

What are some common techniques for factoring double integrals?

One common technique for factoring double integrals is using the Fubini's theorem, which allows you to switch the order of integration. Another technique is using substitution, where you substitute one variable in the integrand with a new variable to make the integral easier to solve. Additionally, you can use symmetry to simplify the integral or split the region into smaller, easier to solve, portions.

What are some real-world applications of integration over a region?

Integration over a region has many real-world applications, such as calculating the volume of a 3D object, determining the mass of an irregularly shaped object, finding the center of mass of an object, and calculating the total amount of a substance in a given region. It is also used in fields like physics, engineering, and economics to solve problems involving rates of change and optimization.

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