In polar coordinates, the average value of a function over a...

In polar coordinates, the average value of a function over a region $R$ (Section 15.3$)$ is given by $$\frac{1}{\text { Area(R) }} \iint_{R} f(r, \theta) r d r d \theta$$ Average distance from interior of disk to center Find the average distance from a point $P(x, y)$ in the disk $x^{2}+y^{2} \leq a^{2}$ to the origin.

In polar coordinates, the average value of a function over a region $R$
(Section 15.3$)$ is given by
$$\frac{1}{\text { Area(R) }} \iint_{R} f(r, \theta) r d r d \theta$$

Average distance from interior of disk to center Find the average distance from a point $P(x, y)$ in the disk $x^{2}+y^{2} \leq a^{2}$ to the
origin.

Key Concepts

Integration Over a Disk

Integration over a disk involves setting up the limits for the radius from 0 to a given radius and the angle from 0 to 2?. This method is especially useful when dealing with symmetric regions like circles, as it simplifies the calculation of area and other integrals.

Polar Coordinates

Polar coordinates are used to describe points in a plane using a distance from the origin and an angle from a reference direction. This coordinate system simplifies integration over circular regions and other radially symmetric areas.

Average Value of a Function Over a Region

The average value of a function over a region is obtained by integrating the function over that region and then dividing by the area (or volume) of the region. This concept is commonly applied in various fields to determine the mean behavior of a function over a given domain.

Jacobian in Polar Coordinates

When converting integrals from Cartesian to polar coordinates, the differential area element changes to r dr d?. The factor 'r' is the Jacobian determinant of the transformation and accounts for the stretching of the coordinate system.

Transcript

00:01 We want to find the average distance from a point on the disk, x squared plus y squared is less than or equal to a squared to the origin.

00:11 So for these set of questions, they tell us that we can find the average by using this equation right here.

00:19 So we're going to need to figure out what our bounded region r is, what the area of this bounded region is, and then some equation that describes the thing we're wanting to find the average of.

00:32 So let's go ahead and start with finding our equation.

00:36 So here it says the average distance to the origin.

00:42 So we're looking for that.

00:44 So remember the origin is the point zero zero.

00:48 So if we were to just take some random point, let's call it x, y.

00:53 Remember we could do x minus zero squared plus y minus zero squared, all square rooted, just by using the distance formula.

01:02 And then that there would simplify down to.

01:05 X squared plus y squared and now we need to convert this to polar coordinates and so we could do x is equal to cosine theta y is equal to um r sine theta but remember x squared plus y squared is really just r squared so we'll end up with square root of r squared and then those would cancel out and we'd have the absolute value of r so if our r ends up being just positive, well, then we can get rid of the absolute value.

01:44 But for right now, we know that this is going to be the absolute value of r for our function.

01:49 All right, now let's go ahead and figure out what our bounded region should be.

01:53 So it says we're stuck in this right here, and that is a circle of radius a on the outside, and then it just has everything shaded in.

02:14 So let's just go ahead and draw that really quickly.

02:16 So it looks something kind of like this.

02:18 So if this is a, this is a.

02:22 We would just have a circle like that.

02:27 And then the entire thing shaded it.

02:31 So on this, our radius would range from 0 to a.

02:39 And then our theta is going to go from 0 to 2 pi, since we're rotating all the way around like that.

02:55 So now we have our bounded region, and then to find the area of this bounded region.

03:08 Also, since it's a circle, we know that that should have an area of pi r squared, and then we can just go ahead and plug in what our radius is, which should be a...

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