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How to find the average squared distance between the points of the unit disk and the point (1,1)

ddd [48]

The unit disk can be parameterized by the function

$\mathbf p(r,\theta)=(r\cos\theta,r\sin\theta)$

where $0\le r\le 1$ and $0\le\theta\le2\pi$ . The squared distance between any point in this region $(x,y)=(r\cos\theta,r\sin\theta)$ and the point (1, 1) is

$(x-1)^2+(y-1)^2=(r\cos\theta-1)^2+(r\sin\theta-1)^2$
$=(r^2\cos^2\theta-2r\cos\theta+1)+(r^2\sin^2\theta-2r\sin\theta+1)$
$=r^2(\cos^2\theta+\sin^2\theta)-2r(\cos\theta-\sin\theta)+2$
$=r^2-2r(\cos\theta-\sin\theta)+2$

The average squared distance is then going to be the ratio of [the sum of all squared distances between every point in the disk and the point (1, 1)] to [the area of the disk], i.e.

$\dfrac{\displaystyle\iint_{x^2+y^2$
$=\dfrac{\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}[r^2-2r(\cos\theta-\sin\theta)+2]r\,\mathrm dr\,\mathrm d\theta}{\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=1}r\,\mathrm dr\,\mathrm d\theta}$
$=\dfrac{\frac{5\pi}2}\pi=\dfrac52$

5 0

3 years ago

please help me with it and dont just put random letters cause i dont know why u do it its not like u can buy a house with these

BartSMP [9]

Answer:

v=432cm

sA=144cm

Step-by-step explanation:

1)))

v=w*d*h

v=3*9*16

v=432cm

2)))

sA=h*w

sA=16*9

sA=144cm

6 0

3 years ago

Read 2 more answers

X - 2y + 2z = -22 x + 3y - z = 03 x + 2y + 3z = -15

scZoUnD [109]

What is your question?

7 0

3 years ago

A. Statistics is the science ofâ€‹ manipulating, reorganizing, and editing information to produce the desired results. Inâ€‹ add

Jlenok [28]

Answer:

The correct definition of statistics is

B. Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw a conclusion and answer questions. In addition, statistics is about providing a measure of confidence in any conclusions.

Explanation:

Statistics is the science of collecting, organizing, summarizing, and analyzing the information and based on this information a conclusion is drawn about the collected information. There are two main divisions of statistics:

1. Descriptive statistics: It deals with providing the summary about the information collected.

2. Inferential statistics: This division deals with using a sample from the population and drawing the conclusion about the population using the sample. It also includes providing a measure of confidence in any conclusions.

8 0

4 years ago

What is 2 X (6+ 2 + 8) -4

VladimirAG [237]

Answer:

The correct answer is 32x - 4

Step-by-step explanation:

1) Simplify 6 + 2 to 8.

$2x(8 + 8) - 4$