# How do you find the area under a polar curve?

## How do you find the area under a polar curve?

To understand the area inside of a polar curve r=f(θ), we start with the area of a slice of pie. If the slice has angle θ and radius r, then it is a fraction θ2π of the entire pie. So its area is θ2ππr2=r22θ.

## What is polar coordinates with example?

Examples of Polar Coordinates: Points in the polar coordinate system with pole 0 and polar axis L . In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60∘) ( 3 , 60 ∘ ) .

How do polar curves work?

A polar curve is a shape constructed using the polar coordinate system. Polar curves are defined by points that are a variable distance from the origin (the pole) depending on the angle measured off the positive x-axis. For example, a cardioid microphone has a pickup-pattern in the shape of a cardioid.

### Why does dA r dr d theta?

So the usual explanation for dA in polar coords is that the area covered by a small angle change is the arc length covered times a small radius “height”. The arc length covered is r * dTheta, and the “height” is dr, so dA is r(dr)(dtheta), where r is the distance away from the center.

### How to calculate an integral in polar coordinates?

In terms of polar coordinates the integral is then, ∬ D e x 2 + y 2 d A = ∫ 2 π 0 ∫ 1 0 r e r 2 d r d θ ∬ D e x 2 + y 2 d A = ∫ 0 2 π ∫ 0 1 r e r 2 d r d θ. Notice that the addition of the r r gives us an integral that we can now do. Here is the work for this integral.

How to enclose an area with polar coordinates?

So, if we use 7 π 6 7 π 6 to 11 π 6 11 π 6 we will not enclose the shaded area, instead we will enclose the bottom most of the three regions. However, if we use the angles − π 6 − π 6 to 7 π 6 7 π 6 we will enclose the area that we’re after. Let’s work a slight modification of the previous example. θ and inside r =2 r = 2 .

#### Can you convert Cartesian coordinates to polar coordinates?

Now, if we’re going to be converting an integral in Cartesian coordinates into an integral in polar coordinates we are going to have to make sure that we’ve also converted all the x x ’s and y y ’s into polar coordinates as well. To do this we’ll need to remember the following conversion formulas,

#### Do you have to use Cartesian coordinates for double integrals?

The problem is that we can’t just convert the dx d x and the dy d y into a dr d r and a dθ d θ . In computing double integrals to this point we have been using the fact that dA= dxdy d A = d x d y and this really does require Cartesian coordinates to use.