Calculate The Double Integral Ye-Xy

What is a Double Integral?

A double integral extends the concept of single integration to two dimensions. It calculates the volume under a surface defined by a function over a region in the xy-plane. The double integral of a function f(x,y) over a region R is written as:

For the function ye-xy, we'll compute the double integral over a rectangular region [a,b] × [c,d].

Calculating the Double Integral ye-xy

To compute the double integral of ye-xy over a rectangular region [a,b] × [c,d], follow these steps:

1. Set up the iterated integral:
   ∫_{x=a}^{x=b} ∫_{y=c}^{y=d} ye^{-xy} dy dx
2. First, integrate with respect to y:
   ∫ ye^{-xy} dy = -e^{-xy} / x + C
3. Evaluate the inner integral from y=c to y=d:
   [-e^{-xy}/x]_{y=c}^{y=d} = -e^{-xd}/x + e^{-xc}/x
4. Now integrate the result with respect to x:
   ∫_{x=a}^{x=b} [e^{-xc}/x - e^{-xd}/x] dx
5. This can be written as:
   ∫_{x=a}^{x=b} [e^{-cx}/x - e^{-dx}/x] dx

The exact value of this integral depends on the specific values of a, b, c, and d. For many cases, it can be expressed in terms of the exponential integral function Ei.

Example Calculation

Let's compute the double integral of ye-xy over the region [0,1] × [0,1].

1. Set up the iterated integral:
   ∫_{x=0}^{x=1} ∫_{y=0}^{y=1} ye^{-xy} dy dx
2. First, integrate with respect to y:
   ∫ ye^{-xy} dy = -e^{-xy}/x + C
3. Evaluate from y=0 to y=1:
   [-e^{-x}/x]_{y=0}^{y=1} = -e^{-1}/x + e^{0}/x = (1 - e^{-1})/x
4. Now integrate with respect to x:
   ∫_{x=0}^{x=1} (1 - e^{-1})/x dx = (1 - e^{-1}) ∫_{x=0}^{x=1} 1/x dx
5. The integral of 1/x from 0 to 1 is ln(x) evaluated from 0 to 1, which is ln(1) - ln(0). Since ln(0) is undefined, we take the limit as x approaches 0 from the right, which is -∞. Therefore, the integral diverges to -∞.

This example shows that the double integral of ye-xy over [0,1] × [0,1] diverges. The behavior of the integral depends heavily on the region of integration.

Interpreting the Result

The result of a double integral represents the volume under the surface defined by the function over the specified region. For ye-xy, the integral's behavior varies significantly based on the region of integration:

• Over finite rectangular regions, the integral may converge or diverge depending on the bounds.
• Over infinite regions, the integral typically diverges due to the exponential decay.
• The integral is undefined at points where the function is infinite, such as at x=0 in the previous example.

When the integral converges, it provides a finite volume measurement. When it diverges, it indicates that the volume is infinite or undefined.

Common Mistakes

When calculating double integrals, common errors include:

• Incorrectly setting up the iterated integral (e.g., reversing the order of integration).
• Miscounting the limits of integration, especially when the region is not rectangular.
• Forgetting to account for singularities in the function (e.g., x=0 in ye-xy).
• Misapplying integration techniques, such as not integrating the constant of integration.

Double-checking each step and verifying with a simpler example can help avoid these mistakes.