Integrating an exponential function over [itex]|x|+|y| \leq 1[/itex]

In summary, the conversation discusses how to integrate e^{-2(x+y)} over a diamond centered around the origin. The person mentions that the region is symmetric over the x-axis and y-axis, but using the symmetry of e^{-x} or e^{-y} is not applicable. They suggest splitting the integral into two parts, with one half of the diamond being integrated over at a time. Additionally, the boundaries for the integrals are discussed, with one choice being to divide the diamond in half by the y-axis. The intervals for integration of the left and right halves are also mentioned.
  • #1
math.geek
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OK, I'm new to multi-variable calculus and I got this question in my exercises that asks me to integrate [itex] e^{-2(x+y)} [/itex] over a diamond that is centered around the origin:

[itex]\int\int_D e^{-2x-2y} dA[/itex]

where [itex]D=\{ (x,y): |x|+|y| \leq 1 \}[/itex]

I know that the region I'm integrating over is symmetric over the x-axis and the y-axis, but [itex] e^{-x} [/itex] or [itex] e^{-y} [/itex] are neither odd nor even to use the symmetry that way.

Obviously, the diamond is symmetric over the axes [itex] x+y [/itex] and [itex] x-y [/itex]. Does this help?
 
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  • #2
Split it into two integrals where one is over one half of the diamond and the other is over the other half!
Then you can write the boundaries of the integrals from [itex] D=\{ (x,y): |x|+|y| \leq 1 \} [/itex] easily.
 
  • #3
Which haves?

Give me some more details please, I'm not looking for a full solution, only an explanation of how you're setting up the integral.
 
  • #4
One choice is dividing the diamond in half by the y axis.Then you have two double integrals,one for the left half and the other for the right one.The interval for the left half integral is [itex] -(x+1) \ to \ x+1 [/itex] for integration w.r.t. y and -1 to 0 for integration w.r.t. x.I think you can figure out the intervals for the right half integral by plotting D.
 

Related to Integrating an exponential function over [itex]|x|+|y| \leq 1[/itex]

What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = a^x, where a is a constant and x is the input value. It is a type of growth or decay function that increases or decreases at a constant rate.

What does it mean to integrate a function?

Integrating a function means finding the area under the curve of the function. It is the reverse process of differentiation, which is finding the slope of a curve at a given point. Integration is used to solve problems involving quantities that are continuously changing over time, such as velocity, position, or growth rate.

How do you integrate an exponential function?

To integrate an exponential function, you can use the power rule, which states that the integral of x^n is equal to (x^(n+1))/(n+1) + C, where C is a constant. For example, the integral of 2^x is (2^(x+1))/(x+1) + C. However, for more complex exponential functions, you may need to use other integration techniques such as substitution or integration by parts.

What is the significance of the absolute value in the given integration problem?

The absolute value in the given integration problem, |x|+|y|≤ 1, indicates that the function is being integrated over a specific region on the coordinate plane. It ensures that both positive and negative values of x and y are included in the integration, resulting in a symmetrical region.

How do you solve for the definite integral of an exponential function over a specific region?

To solve for the definite integral of an exponential function over a specific region, you can use the properties of integration and the fundamental theorem of calculus. First, integrate the function over the given region. Then, plug in the upper and lower limits of integration to find the difference between the two. This will give you the exact value of the definite integral.

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