How to Solve a Double Integral Through a Change of Variables

When dealing with complex double integrals, such as

[ int_0^1 int_0^{1-x} e^{frac{x}{xy}} , dy , dx, ]

one powerful technique is to change the order of integration or use a change of variables. In this article, we'll explore how to solve the given double integral by changing the variables. This approach not only simplifies the problem but also provides a better insight into the solution process.

Introduction

Let's delve into the steps and methods required to solve the given double integral through a change of variables. This technique is particularly useful when the integrand or the region of integration presents challenges. In our case, the integral involves an exponential term with a variable denominator, making it a prime candidate for a change of variables.

Step-by-Step Solution

Step 1: Determine the original region of integration

The original region of integration is defined by the inequalities

( 0 leq x leq 1 ) ( 0 leq y leq 1 - x )

This corresponds to the triangular region under the line ( y 1 - x ) in the first quadrant.

Step 2: Change the order of integration

To change the order of integration, we need to express the region in terms of ( y ) first. The limits for ( y ) will range from 0 to 1. For a fixed ( y ), ( x ) will range from 0 to ( 1 - y ). Thus, we can rewrite the integral as

[ int_0^1 int_0^{1-y} e^{frac{x}{xy}} , dx , dy. ]

Step 3: Simplify the integrand

To simplify the integrand, we use the substitution ( u xy ). This implies that ( du dx ) and ( x frac{u}{y} ). When ( x 0 ), ( u 0 ) and when ( x 1 - y ), ( u y(1 - y) ). The integral becomes

[ int_y^{y(1-y)} e^{frac{u}{u}} , du int_y^{y(1-y)} e^1 , du e int_y^{y(1-y)} 1 , du. ]

After simplifying, we get

[ e int_y^{y(1-y)} du e(y(1-y) - y) e(y - y^2 - y) e(-y^2). ]

Step 4: Evaluate the outer integral

Substitute the result of the inner integral into the outer integral

[ int_0^1 e(-y^2) , dy. ]

This integral can be evaluated using numerical methods such as Simpson's rule or the trapezoidal rule if an exact symbolic solution proves too complex.

Change of Variables

Another method to solve the given integral is to use a change of variables. Consider the transformation ( u xy ) and ( v x - y ). This transformation simplifies the integrand and changes the region of integration. The new region of integration will be triangular as well.

With this change of variables, the Jacobian, which is the determinant of the transformation matrix, is a constant. This simplifies the problem significantly.

Conclusion

In this article, we explored two methods to solve a complex double integral through a change of variables. Whether you choose to change the order of integration or use a change of variables, both techniques provide a powerful tool for solving such problems. Understanding and mastering these methods will greatly enhance your ability to tackle similar integrals in various applications.

Key Takeaways:

Change the order of integration when the region is simple to describe. Use a change of variables to simplify the integrand and region of integration. Utilize the Jacobian to understand the transformation of the region.