Compute the integral by reversing the order of integration

In summary, reversing the order of integration is a technique used in calculus to evaluate a double integral. It involves integrating with respect to the second variable first and then the first variable. This can make it easier to evaluate the integral, especially if one variable is difficult to integrate with respect to. The steps to reversing the order of integration are identifying the limits, rewriting the integral, reversing the order, and evaluating with new limits. However, there are some cases where this technique is not possible, such as when the region is not rectangular or the limits are not functions of the other variable. In these cases, other techniques may need to be used.
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Homework Statement



The problem is in the link.

http://img26.imageshack.us/i/80222189.png/

Homework Equations



None.

The Attempt at a Solution



I did not get far. The only thing I got was the graph. Here is the link to that.

http://img813.imageshack.us/f/graphv.png/

I am suppose to reverse the order of integration. However, I am having trouble finding the limits for x and y.

I spent around 8 hours thinking about this problem -.-

Can anyone please help me with this problem?

Thanks.
 
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Anyone got any tips or hints?
 
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Bump again
 

FAQ: Compute the integral by reversing the order of integration

Can you explain the concept of reversing the order of integration?

Reversing the order of integration is a technique used in calculus to evaluate a double integral. Instead of integrating with respect to one variable and then the other, the order is reversed and the integral is evaluated with respect to the second variable first and then the first variable.

Why would you want to reverse the order of integration?

Reversing the order of integration can sometimes make it easier to evaluate a double integral, especially when the function being integrated is difficult to integrate with respect to one variable but easier with respect to the other.

What are the steps to reversing the order of integration?

The steps to reversing the order of integration are as follows: 1) Identify the limits of integration for the inner integral and the outer integral. 2) Rewrite the integral with the limits of the inner integral as a function of the outer variable. 3) Reverse the order of integration by integrating with respect to the outer variable first and then the inner variable. 4) Evaluate the integral using the new limits of integration.

Can you provide an example of reversing the order of integration?

As an example, consider the double integral of f(x,y) over a region R, where the limits of integration for x are a and b, and the limits for y are g(x) and h(x). The integral can be rewritten as ∫ab∫g(x)h(x)f(x,y)dydx. To reverse the order of integration, we integrate with respect to x first, giving us ∫g(a)h(b)∫abf(x,y)dxdy.

Are there any special cases where reversing the order of integration is not possible?

Yes, there are some cases where reversing the order of integration is not possible. This can occur when the region of integration is not rectangular or when the limits of integration are not constants or functions of the other variable. In these cases, other techniques such as changing the order of integration or using a different coordinate system may need to be used.

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