Integration by Parts of a Double Integral

Thread starter ChabbaBings
Start date Apr 14, 2009
Tags

Double integral Integral Integration Integration by parts parts

In summary, integration by parts of a double integral is a method used to evaluate the integral of a product of two functions over a two-dimensional region. This method involves splitting the integral into two parts and integrating one part while differentiating the other. Its purpose is to simplify the integral and make it easier to compute, especially when the integrand is difficult to integrate individually. However, it has limitations and may not work for all types of integrals. It is most effective when the integrand can be written as a product of two functions and when one of the functions can be integrated while the other can be differentiated. The requirements for using this method include the region of integration being a rectangle or easily breakable into rectangles, and both functions in the

Apr 14, 2009

ChabbaBings

Homework Statement

∫∫xy(x^2+y^2)^(1/2)dydx

over the range 0 to 1 for both x and y.

Homework Equations

I believe that it requires integration by parts.

Any help would be greatly appreciated.

Physics news on Phys.org

Apr 14, 2009

Dick

Science Advisor

Homework Helper

I don't think it requires parts, it's just substitutions. Do the dy with u=y^2. Then the dx with u=x^2. Do the integrations one at a time.

What is integration by parts of a double integral?

Integration by parts of a double integral is a method used to evaluate the integral of a product of two functions over a two-dimensional region. It is an extension of the integration by parts method for single integrals.

How is integration by parts of a double integral applied?

To apply integration by parts of a double integral, the integral is rewritten in the form of a product of two functions, and then the integration by parts formula is used. This involves splitting the integral into two parts and integrating one part while differentiating the other.

What is the purpose of using integration by parts of a double integral?

The purpose of using integration by parts of a double integral is to simplify the integral and make it easier to compute. This method is especially useful when the integrand is a product of two functions that are difficult to integrate individually.

What are the requirements for using integration by parts of a double integral?

To use integration by parts of a double integral, the region of integration must be a rectangle or a region that can be easily broken down into rectangles. Additionally, both functions in the integrand must have continuous and differentiable partial derivatives.

Are there any limitations to using integration by parts of a double integral?

Yes, integration by parts of a double integral may not always work for all types of integrals. It is most effective when the integrand can be written as a product of two functions, and when one of the functions can be integrated while the other can be differentiated. In some cases, other methods of integration may be more appropriate.