Double Integral - Polar Coordinates

In summary, the task is to evaluate an integral by changing to polar coordinates. The given integral involves finding the area under the curve e^-(x^2+y^2) within certain boundaries. To solve this, the intervals need to be changed to polar coordinates, with the substitution of r*drd\phi for dxdy. The exact expression may vary depending on the shape of the domain.
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Homework Statement



Evaluate by changing to polar coordinates

Homework Equations



Can't figure out how to make the integral stop after the sqrt(9-x^2)
[tex]\int_0^\frac{3}{\sqrt(2)} \int_x^{\sqrt(9-x^2)} e^-(x^2+y^2) dy dx[/tex]

The Attempt at a Solution



I'm not sure where to really start on this one. I know it will end up being e^-r^2 but beyond that I'm not sure.
 
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  • #2
You'll have to express dxdy in other variables, and the intervals have to be changed.
 
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How can I change them to polar coordinates?
 
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  • #4
First of all: Draw a figure in the xy-plane with to see what shape the domain yields (probably something like a circle sector). Then you should be able to figure out what values you should give [tex]r[/tex] and [tex]\phi[/tex]. You usually substitute [tex]r*drd\phi[/tex] for [tex]dxdy[/tex] when using polar coordinates. However, the exact expression depends on what shape the domain yields.
 

What is a double integral in polar coordinates?

A double integral in polar coordinates is a type of integral used to calculate the volume or area of a region in two-dimensional polar coordinates. It involves integrating over a region described by polar coordinates, with the integrand given in terms of those coordinates.

What is the difference between a double integral in Cartesian coordinates and polar coordinates?

The main difference between a double integral in Cartesian coordinates and polar coordinates is the way the region is described. In Cartesian coordinates, the region is described by rectangular coordinates (x and y), while in polar coordinates, the region is described by radial distance (r) and angular position (θ). This can make solving certain integrals more convenient in one coordinate system over the other.

How do you convert a double integral from Cartesian coordinates to polar coordinates?

To convert a double integral from Cartesian coordinates to polar coordinates, you can use the Jacobian transformation. This involves substituting the integrand and limits of integration with their corresponding expressions in polar coordinates. The expression for the Jacobian can be found by taking the determinant of the transformation matrix.

What are some applications of double integrals in polar coordinates?

Double integrals in polar coordinates have many applications in science and engineering. They are commonly used to calculate the volume and surface area of three-dimensional objects, as well as the area of regions in two-dimensional space. They are also used in physics to calculate the center of mass and moments of inertia of objects with circular symmetry.

What are some common techniques for solving double integrals in polar coordinates?

There are several techniques that can be used to solve double integrals in polar coordinates. These include using polar symmetry (where the integrand has symmetry with respect to the origin), splitting the region of integration into simpler sub-regions, and using trigonometric identities to simplify the integrand. It is also helpful to visualize the region of integration and choose an appropriate order of integration (i.e. integrating with respect to r first or θ first).

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