Double Integral Homework: Evaluate & Change Order if Necessary

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In summary, when evaluating a double integral, it is sometimes necessary to change the order of integration in order to solve the integral more easily. This can be done when it is convenient or when a given order of integration is not feasible.
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Homework Statement



Evaluate the following double integral. Change order of integration if necessary.

[tex]\int^{1}_{0} \int^{x}_{0} x^2 sin(\Pi x y) dy dx[/tex]

Homework Equations





The Attempt at a Solution



[tex]\int^{1}_{0} \int^{x}_{0} x^2 sin(\Pi x y) dy dx = -\frac{1}{\Pi}\int^{1}_{0} x cos(\Pi x^2 ) dx[/tex]

Let u = x^2 and du = 2x dx

[tex] - \frac{1}{2 \Pi} \int^{1}_{0} cos (\Pi u) du = -\frac{1}{2 \Pi} \frac{sin (\Pi x^2 )}{\Pi} |^{1}_{0} = - \frac{sin( \Pi)}{2 \Pi^2} = 0[/tex]

but that's wrong. Anyone catch my mistake?

I was also wondering when I'm supposed to change the order of integration. Thanks.
 
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  • #2
Hint: What is the value of [tex]\cos(0)[/tex]?
 
  • #3
cos(0) = 1

I see what you meant, let me try it
 
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  • #4
I was referring to when you integrated the sine with respect to y, and received a cosine. The lower integration limit is 0 so it cos(0) should not vanish.



Edit: I see you discovered what I meant. It took 20+ minutes for this computer to load my reply!
 
  • #5
AssyriaQ said:
I was referring to when you integrated the sine with respect to y, and receive a cosine. The lower integration limit is 0 so it cos(0) should not vanish.

Took me a little while, but I got it. Thank you.

I was just wondering when the right time to change the order of integration is, since we never covered it in class.
 
  • #6
When it is convenient. Certainly if you can't do a double integral in a given order you should try changing the order.
 
  • #7
HallsofIvy said:
When it is convenient. Certainly if you can't do a double integral in a given order you should try changing the order.

That certainly makes sense. Thank You.
 

Related to Double Integral Homework: Evaluate & Change Order if Necessary

1. What is a double integral?

A double integral is a type of mathematical calculation that involves integrating a function of two variables over a certain region in a two-dimensional space. It is represented by the symbol ∬ and is used to find the volume under a surface or the area between two surfaces.

2. How do you evaluate a double integral?

To evaluate a double integral, you first need to determine the limits of integration for both variables. Then, you can use a variety of methods such as the rectangular, polar, or cylindrical coordinates to solve the integral. You can also use the Fubini's theorem to change the order of integration if necessary.

3. What is the purpose of changing the order of integration?

Changing the order of integration can make it easier to evaluate a double integral. In some cases, it may be simpler to integrate with respect to one variable first and then the other. This can also help to visualize the region of integration and make the calculations more manageable.

4. What are the common mistakes to avoid when solving double integrals?

Some common mistakes to avoid when solving double integrals include forgetting to change the limits of integration when changing the order, not simplifying the integrand before integrating, and not accounting for the correct variable when using a change of variables.

5. What are some applications of double integrals in real life?

Double integrals have various applications in fields such as physics, engineering, and economics. They can be used to calculate the center of mass of an object, the moments of inertia of a rotating object, and the volume of a solid with varying density. They are also used in economics to calculate the expected value of a function with two variables.

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