5

Consider the following integral (r+2)drAfter making the trigonometric substitution becomes3sin 0the integral ahove9cos? 0 d09 sin" 0 do9 sin € do27sin? 0...

Question

Consider the following integral (r+2)drAfter making the trigonometric substitution becomes3sin 0the integral ahove9cos? 0 d09 sin" 0 do9 sin € do27sin? 0 dO~9cos? 0 d09 _ 9cos? 0 d0Submi A15t=

Consider the following integral (r+2)dr After making the trigonometric substitution becomes 3sin 0 the integral ahove 9cos? 0 d0 9 sin" 0 do 9 sin € do 27sin? 0 dO ~9cos? 0 d0 9 _ 9cos? 0 d0 Submi A15t=



Answers

In Exercises 9-36, evaluate the definite integral. Use a graphing
utility to verify your result.
$$\int_{0}^{\pi / 4} \frac{1-\sin ^{2} \theta}{\cos ^{2} \theta} d \theta$$

Section 84 Problem number eight. So we're dealing with in Arosa Require a trick substitution. Um so integral 0 to 3 halves DX nine months x squared to the three halves power. So the substitution to make in this case is to let x equal three sign of Fada. Then it follows that D X would be equal to three co sign of data. And then let's work on nine minus X squared to the three halves that's going to be nine and that when you square X is going what nine sine squared. So this is gonna end up being 91 minus the sine squared of theta to the three halves. So this is nine to the three halves co sine squared of theta to the three halves. So this ends up being what 27 co signed to the Cube power of data. So when now let's just change the limits of integration. When X is equal to zero, we have to look right here. That means that the sign of zero that means that zero is equal to three sign of data, which tells me that fate is equal to zero when X is equal to three halves. That tells me three halves is equal to three scientific data. So sign of data is 1/2. So that is equal to pi over six. So we now transform this integral to the integral in terms of theta. So it's the integral from zero two pi over six and then you're going to have one over. So this term right here, 27 co sign cubed of Fada and then DX is which you see right here. Three coastline of Fada. So three co signer Theta detainee A little bit clear. So this becomes the integral from zero two pi over six of 3/27. That's just one night and then you're going to have one over co sine squared of theta d theta. So this is 1/9 the integral from zero two pi over 61 over the coastline square that's just seeking squared of Fada d theta. You know, the anti derivative of seeking squared is tangent. So this is 1/9 the tangent of Fada, evaluated from zero two pi over six. So this if you substitute pi over six, you're going to get 1/9 the tangent pi over six one night tangent of zero. I like the zero because that was easier to evaluate. Tangent a pi over six It's going to be the Y over acts of perverse six, so that's gonna be 1/9 and then one over the square root of three minus zero. So this is 1/9 square root of three. If you rationalise that, you get the square to 3/27 as the final answer again. The key here is recognizing the format when you see something that is in the form of a squared minus X squared that tells you that it's a sign substitution for your trick in a room and then make sure you substitute correctly for DX. Um, performed the operation and you should be good to go at that point is trying to take care of all the algebra, trig and arithmetic that could get in the way

Section 84 Problem number eight. So we're dealing with in Arosa Require a trick substitution. Um so integral 0 to 3 halves DX nine months x squared to the three halves power. So the substitution to make in this case is to let x equal three sign of Fada. Then it follows that D X would be equal to three co sign of data. And then let's work on nine minus X squared to the three halves that's going to be nine and that when you square X is going what nine sine squared. So this is gonna end up being 91 minus the sine squared of theta to the three halves. So this is nine to the three halves co sine squared of theta to the three halves. So this ends up being what 27 co signed to the Cube power of data. So when now let's just change the limits of integration. When X is equal to zero, we have to look right here. That means that the sign of zero that means that zero is equal to three sign of data, which tells me that fate is equal to zero when X is equal to three halves. That tells me three halves is equal to three scientific data. So sign of data is 1/2. So that is equal to pi over six. So we now transform this integral to the integral in terms of theta. So it's the integral from zero two pi over six and then you're going to have one over. So this term right here, 27 co sign cubed of Fada and then DX is which you see right here. Three coastline of Fada. So three co signer Theta detainee A little bit clear. So this becomes the integral from zero two pi over six of 3/27. That's just one night and then you're going to have one over co sine squared of theta d theta. So this is 1/9 the integral from zero two pi over 61 over the coastline square that's just seeking squared of Fada d theta. You know, the anti derivative of seeking squared is tangent. So this is 1/9 the tangent of Fada, evaluated from zero two pi over six. So this if you substitute pi over six, you're going to get 1/9 the tangent pi over six one night tangent of zero. I like the zero because that was easier to evaluate. Tangent a pi over six It's going to be the Y over acts of perverse six, so that's gonna be 1/9 and then one over the square root of three minus zero. So this is 1/9 square root of three. If you rationalise that, you get the square to 3/27 as the final answer again. The key here is recognizing the format when you see something that is in the form of a squared minus X squared that tells you that it's a sign substitution for your trick in a room and then make sure you substitute correctly for DX. Um, performed the operation and you should be good to go at that point is trying to take care of all the algebra, trig and arithmetic that could get in the way

Problem. Number 58 we can use either Formula 75 or 70 six. Ah, here. It would make more sense to reduce sine squared to theaters. So we eliminated. So we will use a former 75. Now using A is equal to two and then is equal to two and M is equal to three, and X is equal to that theater. To is the answer for the information you need to sign. Ah to minus one to theater. Cool sign three plus one toe data over 23 plus two, plus Tu minus 1/3, plus two invigoration off sign. Tu minus two Toe leader. Cool sign. Three toe data defeat, which is equal to negative sign to theater. Cool sign for toe theater over 10 plus 1/5. Integration off Cool sign for three to a fate. The fate. Now, by applying formula, we're blind Formula A 68 for the remaining integration with any deeper to three, and A is equal to two for this integration so we can get the the answer. Forties integration is equal to course. I in a square toe later, sign toe later over six plus 2/3 Integration Off Cool Sign. Who would later be Fator, which is equal to 1/3 sign toe data plus C. So substituting in the original integration so is equal to a negative sign to Faith. Cool's time for for toe data over 10 plus 1/5. Cool sign. It's square to think there signed to a theater over streaks plus 1/3. So I, through a faker plus plus C so Dr Simplification, we can reach that. It's 1/30. Science through theater. Negative three. Cool sign. Our four toe data was cool Sign squared Toe data plus two plus I see.

I have an interesting problem, which is a problem. Number 20 off chapter 8.3. So we have this, uh, Inter or where the entering this signed her for powered y times. Course, I swear. Now if he so in this problem, if he attempted Teo the nave substitution, that's you. Because sign off. Why or your course? Course I Why, then this either of the substitution isn't going to make the problem. And the easier you can attempt, I can attempt at ASU. Nice practice. So I'll just say this two will not work. So when When the first attempt, there's a warrant. What? Redus. You know, we try to manipulate Tio inte Grint into a nicer looking function. So it's going to do that well, whenever there is powers of trip functions involved, they did. The the logical thing to do is try to convert him into treat functions with bigger angles but no powers. So, for example, and also when you see a sign of Fei Carson, why must work together? The first thing that they should come to your mind the sign off. Why two icicle too to co sign. Why sign while right so using this week too, right? The Inter as era to pie. Now I will separate two powers off the four and then the sign. So just keep them here and then they remain. There is science. Where? Y times course I squared. Why? Which is huh? Sine squared off to y defined by before using Teo doubling or sign? For now, this looks a little better because it doesn't have anyone for power. And again, ricin, Do we continue and more actually, because we have the sine squared of why here, which we can convert into co sign of two. Why? Using the double angle cosign form. So this would be zero to pie and one liners cur Sign off two Why Over two Time's sign Sward to our work for you. Now, since this's difference off two things, we can separate them and then to each of dental or separate surveys Er tau pi I have the first term would be simply signed squared off to idealize minus one over eight, Sir. To pie um co sign off to why times sine squared off too. Why do you want now? Let's steal this internal. Lost this, um red. This wet into role and deployment blew into separate. So the red into role has our geologist called by right and it's I have 0 to 1 pie. And so this is now Sign off squirt of two. Why? Which is really just an application of the doubling of formally one more time. So I have one over eight Sarah to pie one minus co Sign off to I over to do y. And then and then when you do this, it's going to be one hour 16 throughout a pie one D y find us nice one over 16. It's hard to pie. Course I knw off to whine D'oh! And this will be a pie over 16 actually, um, so So when you Yes, because So this is my empire of 60 minus. Well, technically, there's one of our 32. Sign off to Why that dessert pie? Because sign of two Y is a purely function with pureed pie. It's just pie over 16. Sometimes that's a very useful thing to remember when you're doing with this kind of intervals, and it's too deep blue integration on the next page. So they're blue and the blue is 18 course I've to our times. So 18 course I kn they're the pike or sign off to I time sign squared two. You are now here. We can again convert this into, um so you convert the second part. That sign of squared of two I used a doubling of formal again, but here, actually, the U substitution will make things simpler. So basically, feed to you equal. Sign off to why they will have deal. Sequel to two co sign you two are doing so this whole thing is going to be able to 18 Oh, 18 course I have to. Why? So that's going to be two, You square and to you. But what's interesting is is here that when you play in the boundaries here, well, why it's gone from zero to high, which means use gray from zero 20 So this is entering from 0 to 0, and that's just sirrah. Now, I know this seems weird, but if you look at it closely, it section of excess because both of these coast functions co sign off to why and sign of two. I have periods off have period off. Hi, because there's there's the factors to in it. So we know that for General Xcor, sign of X is equal to co sign explains to pie for any number expert. When you have co sign two X here, then it's not there. Now the period is half because two times we apply in when you change the argument by spite simply why pie, this is co sign off to X class too high and course einer to X. So what this means is that whenever you have a periodic function and that you integrate the function over the same period, resulting in teruel is zero. And this is just a special case off there. That's just something very, very useful fact to remember. So finally, to some of our into its then well, the red into aspire six. Then went rollers flew into a wall or serious. So the answer to our problem here The answer iss Hi. Over 6 10 Now, this was a very nice little challenging exercise because we used, um we manipulate the interference first, and there we used doubling the formulas And then we even used, um, interesting facts about poor Yorick functions integrating over a period into reading over internal AFF length, which is whose length is, say, Mr Pugh. So began the final answer. Here's pie or 60.


Similar Solved Questions

5 answers
3. a. Determine the <100> family of directions in the cubic system, andb. Sketch the members of this family:
3. a. Determine the <100> family of directions in the cubic system, and b. Sketch the members of this family:...
5 answers
2 3 11,a. Solve the Inequalitles: <- r+5 r-2
2 3 11,a. Solve the Inequalitles: <- r+5 r-2...
5 answers
Sketch the surface given by the function f (,y) = 3r2
Sketch the surface given by the function f (,y) = 3r2...
5 answers
Rate {f the 'vears: first in the stream ! income contingous [ produced total income E Find % Raou , 6Q02" {low I5 |()=
rate {f the 'vears: first in the stream ! income contingous [ produced total income E Find % Raou , 6Q02" {low I5 |()=...
5 answers
(a) The distance to star is approximately 7.37 x 10 18 yearsIf this star were to burn out today, in how many years would we see it disappear?How long does it take sunlight to reach Earth? minutes(c) How long does it take for microwave radar signal to travel from Earth to the Moon and back? (The distance from Earth to the Moon is 3.84 105 km.)
(a) The distance to star is approximately 7.37 x 10 18 years If this star were to burn out today, in how many years would we see it disappear? How long does it take sunlight to reach Earth? minutes (c) How long does it take for microwave radar signal to travel from Earth to the Moon and back? (The d...
5 answers
Compute the vector Jinc integral 2 dy | ydc, where C is the part of the curve y = 14 from (0,0) to (1,1) , oriented accordingly (i.e. the choscn oricntation is (rom (0,0) to (1,1).)
Compute the vector Jinc integral 2 dy | ydc, where C is the part of the curve y = 14 from (0,0) to (1,1) , oriented accordingly (i.e. the choscn oricntation is (rom (0,0) to (1,1).)...
5 answers
Find the polynomial that factors to $(6 m-5)(2 m-3)$.
Find the polynomial that factors to $(6 m-5)(2 m-3)$....
1 answers
Using the method of Sec. $10.2 \mathrm{C}$, solve Prob. 10.37.
Using the method of Sec. $10.2 \mathrm{C}$, solve Prob. 10.37....
5 answers
Click Submit to complete this assessmentQuestion 28Which of the following is the polar equation for the given curve_O r = 3 sin 50 0 r = 5 sin 30 O r = 5 + c0530 0 r = 3COs 50Click Submit to complete this assessment:
Click Submit to complete this assessment Question 28 Which of the following is the polar equation for the given curve_ O r = 3 sin 50 0 r = 5 sin 30 O r = 5 + c0530 0 r = 3COs 50 Click Submit to complete this assessment:...
1 answers
Graph each function over a two-period interval. $$y=1-\frac{2}{3} \sin \frac{3}{4} x$$
Graph each function over a two-period interval. $$y=1-\frac{2}{3} \sin \frac{3}{4} x$$...
5 answers
Find a unit vector in the direction of V= 2i-3j-4k
find a unit vector in the direction of V= 2i-3j-4k...
5 answers
20 points; 10 points eachTo make steam, you add 5.60 X 105 J of heat to 0.220 kg of water at an initial temperature of 50.0 %€. Find the final temperature of the steam: (Need t0 draw temperalure versus-heat-added curve) (Cwater 4186 K9xc' = Ly= 22.6 x 105_ Csteam 2010 kaxc What if energy added is 2.8 x 105/. what would be final state of the matter? What is final temperature?
20 points; 10 points each To make steam, you add 5.60 X 105 J of heat to 0.220 kg of water at an initial temperature of 50.0 %€. Find the final temperature of the steam: (Need t0 draw temperalure versus-heat-added curve) (Cwater 4186 K9xc' = Ly= 22.6 x 105_ Csteam 2010 kaxc What if energy...
5 answers
Balance each equation by inserting coefficients as needed.equation [: KHFzKF + Hz + Fzequation 2: NzHaNH, +N2
Balance each equation by inserting coefficients as needed. equation [: KHFz KF + Hz + Fz equation 2: NzHa NH, +N2...
5 answers
Use the following inforrration answ€ QuestionParekeet Feather colour in parakeets controledibuto genes. Blue colour (B) dominani absence color (b1. Yellow colour (Y)E cominani over dbxnce colcuc Wtn allele and allele are present; green parakeet is producedWnat the percent probabllity Obtainino yeilov blue parakeet when terozygous green parakeet crossed with white parakeet?Shoiywork Record your answerwhole numbe roundeoone Occima placeS Beead
Use the following inforrration answ€ Question Parekeet Feather colour in parakeets controledibuto genes. Blue colour (B) dominani absence color (b1. Yellow colour (Y)E cominani over dbxnce colcuc Wtn allele and allele are present; green parakeet is produced Wnat the percent probabllity Obtaini...
5 answers
12, cos?0 + 2cos0 -3 =
12, cos?0 + 2cos0 -3 =...
5 answers
Write an acccptable IUPAC Inmmin for the compound below: (Only systemalic nanes nor common rcictcncc, names art accepted by this qucstion ) Kccp thc IrThc IUPAC namc
Write an acccptable IUPAC Inmmin for the compound below: (Only systemalic nanes nor common rcictcncc, names art accepted by this qucstion ) Kccp thc Ir Thc IUPAC namc...

-- 0.021920--