# Surface Integrals in Polar Coordinates

• DrGlove
In summary, the problem involves finding the area cut from the surface z = 2xy by a cylinder, using rectangular coordinates and then switching to polar coordinates. The conversion from dx*dy to dr*dtheta is important in this problem. The element of area dx*dy can be substituted with r*dr*dtheta in polar coordinates.
DrGlove

## Homework Statement

Find the area cut from the surface z = 2xy by the cylinder x^2 + y^2 = 6.
[Hint: Set up the integral using rectangular coodinates, then switch to polar coordinates.]

## Homework Equations

$$A = \iint \sqrt{{z_x}^2+{z_y}^2+1}dxdy = \iint \sqrt{{r^2}+{r^2}{z_r}^2+{r^2}{z_\theta}^2} dr d\theta$$

## The Attempt at a Solution

$$\begin{gather*} z_x = 2y\\ z_y = 2x\\ \\ A = 4\int_{0}^{\sqrt{6}} \int_{0}^{\sqrt{y^2-6}} \sqrt{4{y^2}+4{x^2}+1} dx dy\\\\ A = 4\int_{0}^{\sqrt{6}} \int_{0}^{\sqrt{y^2-6}} \sqrt{4{r^2}+1} dx dy\\\\ A = \int_{0}^{2\pi} \int_{0}^{\sqrt{6}} \sqrt{4{r^2}+1} dr\frac{\partial x}{\partial r} d\theta \frac{\partial y}{\partial\theta} = \int_{0}^{2\pi} \int_{0}^{\sqrt{6}} r {\cos}^2\theta \sqrt{4{r^2}+1} dr d\theta\\\\ \end{gather*}$$

From there, I can work the problem down, but I'm not sure if my conversion from dx -> dr, or dy -> d theta is even a valid operation. I don't see a ready way to do this problem if I start straight in with polar (cylindrical) coordinates, since the algebra quickly becomes very complex. I guess what I need to know, is how do I appropriately convert dx*dy -> dr*dtheta?

Any help appreciated!

Last edited:
The element of area dx*dy is r*dr*dtheta in polar coordinates. In a problem like this, you don't try to derive it. You just remember it and substitute.

D'oh! Thanks for pointing that out :P

## What is a surface integral in polar coordinates?

A surface integral in polar coordinates is a mathematical concept that is used to calculate the total flux, or flow, of a vector field through a surface that is described using polar coordinates. It involves integrating a function over the surface, taking into account the direction and magnitude of the vector field at each point on the surface.

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

The main difference between these two types of surface integrals is the coordinate system used to describe the surface. In Cartesian coordinates, the surface is described using x, y, and z coordinates, while in polar coordinates, the surface is described using r (distance from the origin), θ (angle from the positive x-axis), and z coordinates.

## How do you set up a surface integral in polar coordinates?

To set up a surface integral in polar coordinates, you first need to parameterize the surface using r and θ. This means expressing the x, y, and z coordinates of the surface in terms of r and θ. Then, the surface integral can be written as an integral of a function of r and θ over a specific region on the r-θ plane.

## What is the significance of the Jacobian when calculating a surface integral in polar coordinates?

The Jacobian is a mathematical term that represents the rate of change of variables in a coordinate system. In the context of surface integrals in polar coordinates, the Jacobian is used to convert the surface integral from Cartesian coordinates to polar coordinates. It is necessary for accurately calculating the total flux through the surface.

## What are some real-world applications of surface integrals in polar coordinates?

Surface integrals in polar coordinates have many applications in physics and engineering. They are used to calculate the flux of electric or magnetic fields through a surface, the flow of fluid through a surface, and the work done by a force on an object moving along a curved path. They are also used in the study of fluid dynamics, electromagnetism, and heat transfer.

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