Multivariable Calculus : Double Integrations on Plane

Example Questions

Example Question #1 : Change Of Variables

Convert the following into spherical coordinates.

Explanation:

In order to convert to spherical coordinates , we need to remember the conversion equations.

Now lets apply this to our problem.

Example Question #2 : Change Of Variables

Evaluate , where  is the trapezoidal region with vertices given by , and ,

using the transformation , and .

Explanation:

The first thing we have to do is figure out the general equations for the lines that create the trapezoid.

Now we have the general equations for out trapezoid, now we need to plug in our transformations into these equations.

So our region is a rectangle given by

Next we need to calculate the Jacobian.

Now we can put the integral together.

Example Question #3 : Change Of Variables

Convert the following into spherical coordinates.

Explanation:

In order to convert to spherical coordinates , we need to remember the conversion equations.

Now lets apply this to our problem.

Example Question #1 : Double Integration

Calculate the following Integral.

Explanation:

Lets deal with the inner integral first.

Now we evaluate this expression in the outer integral.

Example Question #2 : Double Integration

Calculate the following Integral.

Explanation:

Lets deal with the inner integral first.

Now we evaluate this expression in the outer integral.