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Solve the problem: Find the average distance from a polnt P(r; 0) In the region bounded by r=4+5sm @ to the origin; 4 5 # 4...

Question

Solve the problem: Find the average distance from a polnt P(r; 0) In the region bounded by r=4+5sm @ to the origin; 4 5 # 4

Solve the problem: Find the average distance from a polnt P(r; 0) In the region bounded by r=4+5sm @ to the origin; 4 5 # 4



Answers

Compute the average value of the following finctions over the region $R$ $$f(x, y)=4-x-y ; R=\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 2\}$$

In prominent 45. We want to get the average value for the function F of X equals four X plus five through the interval from zero to a. He can use the definite integral to calculate the average value for dysfunction. Boy calculating all evaluating the different Integral for dysfunction for X plus five, the X over the interval from zero to a divided by the length off. This interval, which is a minus zero, equals one divided by him multiplied boy. The integration off full X is four x squared, divided by two plus the integration or +55 x we integrate from zero to a we substitute by X equals a We started by our bones for a squared divided by two plus five b and we substitute minus. We substitute by X equals zero So sued by the lower bone which gives you equals one divided by a but the boy boy toe is good. Last five equals two a plus five and this is the average value for the function four x last five over the buried or the interval from 0 to 8 and this is the final answer. This is the final answer of all property

Our problem 40 on the average squared distance between the origin and the points on the proble Z equals four minus X squared minus y squared for a secretive than zero. So we have Z than we can tell that this is a circle ah x squared plus y squared equals four The radios to so set up the integral for the volume cause integral from 0 to 2 pi in the world from 0 to 2 Four miners are squared are er de theater and simplifying this we get two pi times thio r squared minus our fourth over four integrated from 0 to 2 and that gives us a pie and then for the distance from the origin the point I need to find the average value So we have one over v girl of F the A ID equals one over plug It won over a for a volume and the girl from negative to to to you don't grow from negative four minus X squared four months x squared of X square plus y squared the x t y, and evaluating that and grow end up with a pie Great pie, which is gonna want

Function X Why you go to the four minus x amount of skwy on the ricin on in current u x y and we have the excess between guzzone to Onda. Why is between the turn to vote and Kendari the Aryan Origen you go to Ju attempts to it and country fall Now when did you find a double in the ground? A function f x Now we can put it the x and the y Yeah, extra albums urge you to and Wagner from the virtuous Well, then we get Nico Jew we'll give the outer for now Then we have the why but it you know we have ever been in the X so x will be the variable while been a constant them timely Refer to a fugitive for X minus X squared and a ju minus x y involvement Ondas are too They were getting culture to do it. We would it join some under eight? My last, uh, he'll have to square one is to let us do why And then why are we getting on? The coach is achy to six months to why d why an entirely lifted a six echo June uh, six quiet minus Kwai Square Devalued the antidote to Then we get Nico Children throw minus that Just Quentin country far. So you could You ate and I found the average Joe. You go to the depot into grow Nobody Aria And could you ate out of far on a good Jew?

So I've written down the formula to find the average value of the function F. And it's just equal to one divided by b minus a times the integral from A. To B. Of our function F. Of X. Dx. And so here they would be equal to zero since it's our lower bound and B would be equal to four. So we can plug these values in and plug our function F. Of X, which F. Of X was equal to four times E. To the power of X divided by two. So we can plug these values in for A. B. And F of X. And we'll find our average value for our function. So average is equal to one divided by for minus zero, multiplied by the integral from A. Is equal to 00 to four. Of our function F. Of X, which is equal to four times E. Raised the power of X divided by two dx. So this is equal to 1 4th Times the integral from 0 to 4 Of four times the race the power of x divided by two d. x. And so what we can do is we can take out this constant four and we can multiply it by the 1/4 and we're just left with one. So this is equal to the integral of E. To the X divided by two D. X. from 0 to 4. So what we can do now is you can find this integral. So let's let U equal X divided by two. And then that said to you equal one half. So now our function are integral is from 0 to 4 still. And it's the integral of E. To the U. Multiplied by two. Since the U. Is equal to one half, we have to multiply by two and multiply by D. You. So the integral of each. The use just each of the you. So this is gonna be equal to two times E. To the U. From 0 to 4. And so we can plug back in what U. Is equal to. And it was equal to X divided by two. So the sequel to Time is x divided by two from 0 to four. And so if you plug in X is equal to four, we have this would be equal to two times E squared. And then when X is equal to zero, we just have two times E. To the zero power or minus two. And so this is equal to two times E squared minus one.


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