# Integration of Polar coordinates

• DunWorry
In summary, the process of finding the area in a polar curve involves calculating the area of a small section of the curve using the formula dA = \frac{1}{2}r^{2}dθ and then integrating this over the given limits. This is different from the method used in cartesian coordinates, where the function is simply plugged into the integral. The reason for this difference is that in polar coordinates, an additional step is needed to calculate the area.
DunWorry

## Homework Statement

Find the area in the polar curve r = sin2θ between 0 and $\frac{\pi}{2}$.

The way to do this is to say the area of a tiny bit of this polar curve, dA = $\frac{1}{2}$r$^{2}$dθ

so the integral is just $\frac{1}{2}$$\int$$^{\frac{\pi}{2}}_{0}$(sin2θ)$^{2}$dθ

if we did say a function in cartesian coordinates, eg y=x we just do $\int$x dx. I am confused as to why in polar coordinates we cannot just do the same and say do $\int$sin2θ dθ, I understand how to get the correct integral, but what I am asking is what is wrong with just integrating the polar function like we do in cartesian coordinates? why do we have to say we will take a bit of this sector and integrate it over these limits, for polar function, but for a cartesian one we just plug the function into the integral?

Thanks

In both methods, you skip one step to calculate an area. In cartesian coordinates, this step is trivial (you have to multiply with 1), in polar coordinates, it is not.
See the discussion here, for example.

## 1. What are polar coordinates?

Polar coordinates are a coordinate system used to describe the position of a point in a two-dimensional plane. They consist of a distance from the origin (r) and an angle from a reference line (θ).

## 2. Why is it important to integrate in polar coordinates?

Integrating in polar coordinates allows us to solve problems that cannot be easily solved using rectangular coordinates. It also helps in solving problems involving circular or rotational symmetry.

## 3. What is the formula for converting from rectangular coordinates to polar coordinates?

The formula for converting from rectangular coordinates (x,y) to polar coordinates (r,θ) is: r = √(x² + y²) and θ = arctan(y/x).

## 4. What is the process for integrating in polar coordinates?

The process for integrating in polar coordinates involves converting the integrand from rectangular form to polar form, using appropriate substitutions for the variables, and then evaluating the integral using the polar integration formulas.

## 5. What are some applications of integrating in polar coordinates?

Integrating in polar coordinates has many applications in physics, engineering, and mathematics. It is used to calculate areas and volumes of regions with circular or rotational symmetry, such as the area of a sector or the volume of a sphere. It is also used in calculating work done by a force acting in a circular path, calculating moments of inertia, and solving differential equations involving circular or rotational motion.

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