How Do I Set Up Limits for a Double Integral Over a Triangular Region?

In summary, the conversation discusses how to solve a definite double integral in the area of a triangle with vertexes at (0,0), (1,1), and (0,2). The equation to be solved is the integral of e^(y^2) dy*dx. The conversation suggests defining the limits for the integral using the lines x=y, x=0, and y=-x+2. There are two possible approaches to solving the integral, but one of them cannot be solved. It is suggested that the problem may have been miscopied or mis-stated, but the overall topic is understood and the student is prepared for the upcoming test.
  • #1
ju0020
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Homework Statement


I'm supposed to solve a definite double integral. It's supposed to be in the area of the triangle with vertexes at (0,0), (1,1),(0,2)

Homework Equations


integral of
e^(y^2) * dy*dx

The Attempt at a Solution


First, I need to know the limits of x and y...
So, that triangle is defined by the lines x=y, x=0 and y=-x+2
with that I can define the limits for my integral

(ps. I don't know how to use latex very well so this will look kind of weird, but what's on top is the upper limit and what is under that is the lower limit).

I've tried it in the order [tex]\int[/tex][tex]\stackrel{1}{0}[/tex] [tex]\int[/tex][tex]\stackrel{-x+2}{x}[/tex] e^(y^2)*dy*dx
and [tex]\int[/tex][tex]\stackrel{1}{0}[/tex] [tex]\int[/tex][tex]\stackrel{y}{0}[/tex] e^(y^2)*dx*dy + [tex]\int[/tex][tex]\stackrel{2}{1}[/tex] [tex]\int[/tex][tex]\stackrel{0}{2-y}[/tex] e^(y^2)*dx*dy
But it both cases at some point I can't solve it. For example, in the second option I end up with an integral of 2*e^(y^2) and in the first option I can't even begin to solve that integral


I have a test in a few hours so any help is much appreciated. Thanks
 
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  • #2
The limits on your very last integral are reversed, should be 0 to 2-y. You have the right idea about trying a dxdy integral, and the first one of the dx dy integrals works. You are correct the second one can't be worked. The problem might be miscopied or mis-stated such that the third point should have been (0,1), in which case you would have gotten it. Don't worry about it for the exam, you are good to go on that topic.
 

FAQ: How Do I Set Up Limits for a Double Integral Over a Triangular Region?

1. What is a definite double integral?

A definite double integral is a mathematical tool used to calculate the area under a two-dimensional curve. It is written as ∫∫f(x,y) dA, where f(x,y) is the function being integrated and dA represents the area element.

2. How do I solve a definite double integral?

To solve a definite double integral, you need to follow a specific process. First, determine the limits of integration for both x and y, then evaluate the integral using the appropriate integration techniques (such as substitution or integration by parts).

3. What is the purpose of a definite double integral?

The purpose of a definite double integral is to find the area under a two-dimensional curve. This is useful in many fields, including physics, engineering, and economics.

4. Can I use a calculator to solve a definite double integral?

Yes, you can use a calculator or a computer program to solve a definite double integral. However, it is important to understand the underlying concepts and techniques used in solving the integral.

5. Are there any tips for solving definite double integrals?

Yes, there are a few tips that can make solving definite double integrals easier. These include breaking the integral into smaller parts, using symmetry to simplify the integrand, and practicing with a variety of examples.

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