5

Question 23 (1 point) LeatanFind the average value ot f(=,y) ry over the rectangle with vertices (0,0), (5, 0), (5,3). (0,3).None of these;0 ! $ [ (2ry)dyd = 0% J f...

Question

Question 23 (1 point) LeatanFind the average value ot f(=,y) ry over the rectangle with vertices (0,0), (5, 0), (5,3). (0,3).None of these;0 ! $ [ (2ry)dyd = 0% J f3 (ry)dydr = J5 [ (2ry)dy dr 0 % J8 [3 (2cy)dy dr =

Question 23 (1 point) Leatan Find the average value ot f(=,y) ry over the rectangle with vertices (0,0), (5, 0), (5,3). (0,3). None of these; 0 ! $ [ (2ry)dyd = 0% J f3 (ry)dydr = J5 [ (2ry)dy dr 0 % J8 [3 (2cy)dy dr =



Answers

Find the average value of $ f $ over the given rectangle.

$ f(x, y) = x^2 y $,
$ R $ has vertices $ (-1, 0) $, $ (-1, 5) $, $ (1, 5) $, $ (1, 0) $

Okay, so first, let's look at the rectangle that we're integrating. Over. Uh, negative 10 negative. 15 15 and 10 website. Okay, so we have X going from negative 1 to 1. And why going from 0 to 5. And so the width is too. And what's the width is to and the length iss five. So the area of the rectangle is 10. So the average value of F B the integral I'll be 1/10 the other girl Here are five negative 1 to 1 x squared y dx dy oir. Because this is the volume. This part is the volume underneath X squared y on this rectangle. And so if we divide by the area will get the average value. All right, so we're gonna integrate with respect to X first. Whoops. Mhm. Do you? So we have 1/10 05 Execute over three from negative one toe one. Why do you Why? I mean to move over here a little bit. 1/10 and a girl. 05 one third, minus negative. One third. Why do you I so to 30th 1/15 0 to 5. Why do you Why 1/15? Why Scored over to most two from 05 So 25/30 or 56

So you want to go ahead and find the average value of F of X Y is equal to X squared. Why? Over our rectangle, which is given as courtesies. But we can go ahead and break these down as the following, uh, balance. So then we know where average value is just one over the area of our times double winner role of all over our of the function. So then, um, just looking at this, we know that this is a two by five rectangle. So are a of are just ahead. So then we can go ahead and from getting and plug in our it's a grand. So, um, since they're just both product of X and Y, there's no reason to choose one order over the other it. So I'm just gonna choose why medics. So go ahead and rewrite this with their expounds on the outside and her Why balance on the inside of expert? Why, u y the x. So then I'm just gonna be left with, um the outside integral and evaluating the inside in the world, we get X squared y squared over to from 0 to 5 will close off the outside inner girl. So then we can go ahead and plugging our values and noticed that at zero this all goes away. So we're just left that a white was five. So 25 x squared over two decks so we can go ahead and pull out a 25 to I noticed that 25 2 times 1/10 is just 5/4. So then we just have 5/4 times integral of X squared DX, which is just gonna give us executed over three from negative one. The one. Oh, are I Fort Times 1/3 plus 1/3 or 2/3. So we're average value just ends up being 10/12 or over since so that's the average value of dysfunction over our given letting.

We're gonna find the average value of F on this region, which is a rectangle. And so here's what we're going to dio average value of F equals. We're going to do one over the area of the region times the volume. The reason that works. It's because this is one over length times with and then the volume is a bunch of rectangular prisms which are linked with height. And so when you get done, you get high on the average. Okay, someone put length with height there because the thing isn't really a rectangular prism, but each piece is okay, so this is a rectangle and it's areas four times one so average value 1/4 and a girl I'm going to do what e d y first. So let's put X on the outside. 04 gear toe one. I'm gonna write this way X plus e to the y E to the y d y DX. So I'm gonna make a substitution because there's a neat the white inside this function and we eat the white on the outside. I know that I want to make this X plus e to the y equal to you and then do you x is a constant with respect to y So it's driven of a zero derivative of e to the y Eat the y d y. I like to switch my in points to if, uh, Y equals zero then U equals X plus e to the zero, which is one if why equals one u equals X plus e to the one or x plus e Okay, so this integral becomes 1/4 she heard before X plus one x plus e you to the one half all this stuff right here turns into d'you. Yeah, DX. Okay, so now I'm ready to integrate down a little bit here. So we get 1/4 tier 24 you to the three halves over three halves X plus one x plus e dx. Okay, The two thirds flips around you get 16 04 x plus Z to the three halves minus X plus one to the three halves, the X You know, on this part I'm not going to do, do you substitution because I don't have to. Sorry about that because I have X Plus E. If that was you, then do you would be d X, so I'm all good. I'm just gonna go ahead and a great 1/6 x plus e to the five halves over five. House minus X plus one to the five halves over five halves. I'm sure before so the five halfs flips around. You get 1/15 four plus e to the five halves, minus five to the five halves, minus e to the five house, minus one to the five house, the only simplifying. I can see that you could do there. You could make this one to the five at us one. But everything else is still gonna be all grow. So I'm just gonna leave it like that, Yes.

Hi there in this problem. We have several parts, but in part a were just asked to find the average value of this function F over this rectangle are. So they described it for us. I've drawn it here just to help us see it a little more. And we're used to this may as well get the area are now before we start And since the basis for and the height is also for the area is going to be 16. So we're starting, we want to find the average value of this function F over the rectangle are and we know from this chapter that equals one over the area of our We know that's 16 and multiply that by the double integral over our of the function F. Okay, So we keep the 1 16th. We have to set up our integral. The function f is what they gave us. This looks a little more amusing than some of the other homeworks we've had in his set. So we'll see how this goes. Um Looks like the order shouldn't matter very much. We just have so many separate terms here. So let's do dx first and then dy and X goes from negative to to to and why also goes from negative to to to. Okay. So we start as always from the inside. So we need the anti derivative with respect to X Which is next to the third gives us extra the 4th over four. Why do the third gives us X times Y to the third. Be a little careful here. So we have minus E. To the minus X. It's anti derivative is itself Except that minus X. And the exponent by the chain rule. That should have produced a minus. So we have to change the sign here. Let's take a second. Make sure you agree with the fact that positive E. To the minus X. As the derivative minus E to the minus X. So that's correct. And then -1 becomes a -X. And we're going from -2 to 2. Okay We'll have to plug in some things here and probably will want to calculator at some point in the near future. When you plug in two for all the excess, let's see. Two to the fourth is 16, Which over four is 4 plus. We had two times y cubed Plus E to the -2 minus just too. So that's all that we get from plugging in two. Now we have to plug in -2 for all the exes and see what we get. So negative two to the fourth is still positive 16. So we still get a four from that first term but we get minus to the third. Um We get let's see plus E. Negative -2 is either the positive too And finally we get a plus two right there. So that's all what came from plugging in -2 for all the exes. Okay so simplify a little bit here. So we do have so the four and the -4 do cancel out. We get to y cubed minus a negative two Y cube. So that should be four Y cubed. Get a plus E to the -2 minus E to the positive too. And finally with the constants negative two minus another two. So we should have minus four. Okay. Mhm. Get near the end here need an anti derivative with respect to why. So for y cubed that has anti derivative of exactly why to the 4th. And this here is just one big constant. So it's anti derivative is why times this constant. Okay go ahead and plug things in here. So we'll first plug in two for otherwise and we get 2 to the fourth is 16 plus two. E. To the minus two minus two. You to the second minus students. Four is eight. I'll combine right now. The 16 -8 Is just eight. Right? So the constantly end up with here should be eight. That's what we get by plugging in positive two. Now let's plug in negative too. And see what we get Native to the 4th is still 16. Did we get -2 times? I'll go ahead and expand just like before -2. E. to the -2 plus here. The positive plus to eat a positive to sorry? Plus two times four plus eight. Okay so in the 16 plus the eight This time is 24. Okay, So let's combine and then probably just grab a calculator after this step 8 -24 is -16. Yeah. two U. to the negative too Plus another two years of the negative too. So it's plus four E. To the minus two. Likewise here we had minus two X squared minus another two E squared so minus four E squared multiply that all by 16. And if you want a decimal answer you can just type all this new calculator but the hard part is done so that's the average value for part A. Mhm. Um First though you're just a little alarm it looks I may have a slightly different book that I'm looking at a version of the book. Um Some of you may have had the original function. I'm just looking now online and looks like Some of you may have had this original function being at two. So if so and if you're wondering where I got the three from um we have a different version of the same problem I believe. So if that happened to you, um which might be most of you, let me just go through this, we're not going to do everything cause we really don't have to again just changing this exponent on the y from a three to it to everything still is fine until this step right here. Anti derivative of four Y squared Would have been 4/3 why cubed like that? So let me just, just what I've done everything fairly for everybody. Um It'll still be a mess. You have to do with a calculator but he would have gotten four thirds two cubed is eight, 4/3 times eight. And then plus all this other stuff is the same. This is again those things we can do in a calculator uh minus, I'm not plugging in the -2s. Again This time negative to Cuba is -8. And then we'll get -2 times the same constant. Okay. And then again just I I really wouldn't do too much more algebra at this point. You can just type this last step we have into a calculator if you need the decimal expansion, but this will certainly do the trick. Okay, so let's go on a part B now. And we are asked to plot the points for which F. And for this I'll use the version. I just did because I think this is the more common one. Maybe the more recent version? Um Dysfunction F. So we're asked to graph We're an equal zero. Um You're not expected to do that by hand, I don't think so. You can use some graphing software or as most dot com or whatever is easiest for you and you get something like this. And in our region are which was negative to positive, two for both variables. This is our region are so zooming in. There's a red curve. That's all the points for which dysfunction have here equals zero. All right. Look, finally, we're asked for which points in our one is the function greater than zero and for which points is the function less than zero? Well, In part B, we've already traced all the points where the function equals zero and you can't change from greater To less than zero without going through it. So um if we think of like an entire half or it's not quite half, but everything to the right side of this curve that either must all be above zero or all be below zero because the function is continuous. And uh it can't become can't go from positive to negative without going through zero. So if we just think um really we can just plug in a test case, plug in something like maybe right on the edge here like two. Or actually Since we see the entire graph, we could just put in something way to the rights, imagine X being 100. And if X is huge, we can see right away. This third is actually the third term that's going to dwarf everything else will definitely end up with something positive. So uh F is positive is greater than zero on the right part. For sure. And let's thinking now about the left part, I'll color that in yellow and we can do a similar thing that just test the point. Imagine X being negative. A big negative number something way over here and this time a big negative number cubed is a big negative number, so we'll end up with something negative. So our function is negative on the left half. Yeah. And we are done with this problem. Hopefully that was helpful.


Similar Solved Questions

5 answers
B) The process x(t) is normat with meanautocovafiance e-a/v/ Let(t)dt(a) Find P{ inn / < €}be chosen So that Pln - nl <4} is not Iess than 95? (6) How large should
B) The process x(t) is normat with mean autocovafiance e-a/v/ Let (t)dt (a) Find P{ in n / < €} be chosen So that Pln - nl <4} is not Iess than 95? (6) How large should...
5 answers
The 1 Which type Question 2 Susplococcut 1 Streptococcus Sueptococtur Ropoyaoointen Duenum 1 DtulciL L that Is 1 pncumone 8 1 Most comion 1 1 { common ! 8
The 1 Which type Question 2 Susplococcut 1 Streptococcus Sueptococtur Ropoyaoointen Duenum 1 DtulciL L that Is 1 pncumone 8 1 Most comion 1 1 { common ! 8...
5 answers
Let F = Q(v3 + V7)_ Prove that [F:Q] = 4 List all the ring homomorphisms from F to C that fx Q For each one;, state where it sends V7 and V3+ V7, and whether or not the image of the homomorphism C is equal to F. Is F/Q & normal extension? Identify the group Gal(F/Q) of isomorphisms F + F that fx Q
Let F = Q(v3 + V7)_ Prove that [F:Q] = 4 List all the ring homomorphisms from F to C that fx Q For each one;, state where it sends V7 and V3+ V7, and whether or not the image of the homomorphism C is equal to F. Is F/Q & normal extension? Identify the group Gal(F/Q) of isomorphisms F + F that fx...
5 answers
A chemical laboratory is given the following samples to test for pollutant known to hiave standard deviation of 1.SmglL {48.3.51.5.47.7. 52.5,47.4. 50.3} mg-L Compute the 95% confidence interval O tle Qean
A chemical laboratory is given the following samples to test for pollutant known to hiave standard deviation of 1.SmglL {48.3.51.5.47.7. 52.5,47.4. 50.3} mg-L Compute the 95% confidence interval O tle Qean...
5 answers
Pt) Solve the following initial-boundary-value problem:Ut Au = 0 u(c,y,0) = 1 u(0, %,t) u(T,9,t) = u(t, ( 0,+) u(c, T,t) = 0u(z,9,+) = Cn,m Tl=[ TIL = Wherecn,m
pt) Solve the following initial-boundary-value problem: Ut Au = 0 u(c,y,0) = 1 u(0, %,t) u(T,9,t) = u(t, ( 0,+) u(c, T,t) = 0 u(z,9,+) = Cn,m Tl=[ TIL = Where cn,m...
5 answers
Evaluate the integralarctan _dr. 2x + 2
Evaluate the integral arctan _ dr. 2x + 2...
5 answers
Determine whether the series converges or = diver +5 " = [ 45
Determine whether the series converges or = diver +5 " = [ 45...
5 answers
(11) Iin (V1ezt+3 4) (Fint: Rationalize the numerator )(12) lim (VIr" + <- J2)(13) lin (& Vr -I+5) Ic
(11) Iin (V1ezt+3 4) (Fint: Rationalize the numerator ) (12) lim (VIr" + <- J2) (13) lin (& Vr -I+5) Ic...
5 answers
[~/1 Points]DETAILSLARCALCET7 5.7.010_appropriate. Use for the constant integration ) Find the Indefinite integral: (Remember use absolute values where
[~/1 Points] DETAILS LARCALCET7 5.7.010_ appropriate. Use for the constant integration ) Find the Indefinite integral: (Remember use absolute values where...
5 answers
21 Arver cutrchl (daection oLYaris) ccrts torce 0t 120 onYoui enllbont (m strong wind comes Up wllh 3 force 0f 186 Non your bot In Uhe tlirectlon ol 38 NLLlhe_XaXlsWhat Is the f1ni acceleraton on Yout bojt} Accelernon Yeiol Bive the direction well rnneniudg
21 Arver cutrchl (daection oLYaris) ccrts torce 0t 120 onYoui enllbont (m strong wind comes Up wllh 3 force 0f 186 Non your bot In Uhe tlirectlon ol 38 NLLlhe_XaXlsWhat Is the f1ni acceleraton on Yout bojt} Accelernon Yeiol Bive the direction well rnneniudg...
5 answers
For the following functions, find: Critical points Local maximum and minimum lues Points of inflection Intervals of increase and decrease Intervals of concavity f (x) 6r2 9(x) Fr' 3x" h(x) k(r)
For the following functions, find: Critical points Local maximum and minimum lues Points of inflection Intervals of increase and decrease Intervals of concavity f (x) 6r2 9(x) Fr' 3x" h(x) k(r)...
1 answers
The maximum percentage error in the measurement of the surface area of this cube by the above method is (a) $\frac{1}{60} \%$ (b) $\frac{1}{6} \%$ (c) $\frac{1}{600} \%$ (d) $\frac{1}{6000} \%$
The maximum percentage error in the measurement of the surface area of this cube by the above method is (a) $\frac{1}{60} \%$ (b) $\frac{1}{6} \%$ (c) $\frac{1}{600} \%$ (d) $\frac{1}{6000} \%$...
5 answers
C) Fehling'$ reagent is used to distinguish between aldehydes and ketones: Why? write the chemical equations
C) Fehling'$ reagent is used to distinguish between aldehydes and ketones: Why? write the chemical equations...
5 answers
A first order reaction has activation energy: Ea = 3.93 kJ-mol-' and has rate constant: k = 0.727 $ ! at a temperature of 22 %C. What will be the reaction rate constant (in units of s ') if the temperature is lowered to 8 "C?
A first order reaction has activation energy: Ea = 3.93 kJ-mol-' and has rate constant: k = 0.727 $ ! at a temperature of 22 %C. What will be the reaction rate constant (in units of s ') if the temperature is lowered to 8 "C?...
5 answers
Explain in your own word which aspect of SCIENCE do you believeis most important to modern society, and why is it important
Explain in your own word which aspect of SCIENCE do you believe is most important to modern society, and why is it important...
5 answers
Did you see any DNA strands?If so, what did they look like? If not, please explain whyyou think it did not work.
Did you see any DNA strands? If so, what did they look like? If not, please explain why you think it did not work....
5 answers
Bx + C after clearing fractions t0 decompose the equalion Sx-1=A (sx2 (+6)(5x2 +1) X+6 Sx2 _(Bx + C)x + 6) results If x = 6, what is the value of A?The value of A is (Simplify your answer; Type an integer or simplified fraction;)
Bx + C after clearing fractions t0 decompose the equalion Sx-1=A (sx2 (+6)(5x2 +1) X+6 Sx2 _ (Bx + C)x + 6) results If x = 6, what is the value of A? The value of A is (Simplify your answer; Type an integer or simplified fraction;)...

-- 0.017624--