After the transformation, the resulting figure is a circle of radius 1, centered at the origin. If r ranges from 0 to 1, what needs to be the range for ##\theta## to sweep out the whole circle?rashida564 said:
Right.rashida564 said:
A change of variable in a double integral is a technique used to simplify the integration of a function over a region in two variables. It involves substituting a new set of variables in place of the original variables, which allows for a more manageable integral to be evaluated.
A change of variable can be useful in double integrals because it can help to simplify the integrand and make it easier to evaluate the integral. It can also allow for the use of different coordinate systems, which may be more suitable for certain types of functions or regions.
The choice of variables for a change of variable in a double integral depends on the shape of the region being integrated over and the function being integrated. Generally, the goal is to find a substitution that will transform the region into a simpler shape, such as a rectangle or a circle, and make the integrand easier to work with.
The steps for performing a change of variable in a double integral are as follows: 1) Determine the new variables to be used and the transformation equations that relate them to the original variables. 2) Express the original integral in terms of the new variables. 3) Determine the limits of integration for the new variables by transforming the limits for the original variables. 4) Solve the transformed integral using the new limits and the new variables.
In theory, a change of variable can be applied to any double integral. However, in practice, it may not always lead to a simpler integral or be possible to perform. It is important to carefully consider the region and function involved before attempting to use a change of variable.
