5

(1 pt) Suppose the solid W in the figure is one-quarter of a circular cylinder of height 6 and radius 8 centered about the z-axis in the first octant: Find the limi...

Question

(1 pt) Suppose the solid W in the figure is one-quarter of a circular cylinder of height 6 and radius 8 centered about the z-axis in the first octant: Find the limits of integration for an iterated integral of the form fdV = I" "[' f(r,0,2) dz rdr d0.C=D =[Enable Java t0 make this image interactive]E=F =If necessary; enter 0 as theta(Drag to rotate)

(1 pt) Suppose the solid W in the figure is one-quarter of a circular cylinder of height 6 and radius 8 centered about the z-axis in the first octant: Find the limits of integration for an iterated integral of the form fdV = I" "[' f(r,0,2) dz rdr d0. C= D = [Enable Java t0 make this image interactive] E= F = If necessary; enter 0 as theta (Drag to rotate)



Answers

Let $R$ be the region bounded by the curve $y=\cos ^{-1} x$ and the $x$ -axis on $[0,1] .$ A solid of revolution is obtained by revolving $R$ about the $y$ -axis (see figures). a. Find an expression for the radius of a cross section of the solid at a point $y$ in $[0, \pi / 2]$ b. Find an expression for the area $A(y)$ of a cross section of the solid at a point $y$ in $[0, \pi / 2]$ c. Write an integral for the volume of the solid. (IMAGE CAN'T COPY)

Were given an integral and were asked to sketch the solid, whose volume is given by the central into evaluating Integral. This is the interval from year to to and to grow from 0 to 2 pi into grow from zero r of our data D r. Well, we'll see if the region of integration this is given in cylindrical coordinates has the set e of all triples, our fate a Z such that data lies between zero and two pi. Our lives between zero and two and Z lies between zero and are so this represents the solid region enclosed by the cylinder. In fact, the circular cylinder r equals two founded a boat by the cone Z equals R and bounded below by the X Y plane. So with this information, let's stench the graph of the region of integration. Remember exploring Z axis? Okay. Now, as I said, this isn't close by the circular cylinder of radius too. Access the Z axis. This looks something like this. Yeah, And this cylinder this under sex with cones equals R in the plane. Z equals two as a circle 19 points. You get the cylinder men inside the cylinder will draw the cones enclosed bar. This is a vertex at Yeah, origin. And it connects to the circle above like this. And so this is what our regional integration looks like. Next, let's evaluate this. Integral the integral from 0 to 2, integral from 0 to 2. Pi him to grow from zero to our of our easy data D r. Well, if you take anti derivatives with respect dizzy, this isn't to grow from zero to integral from zero to pi of our Z from Z equals zero to Z equals R data yard. This is integral from zero to integrate from zero to pi uh, r squared debated er by Levine's theory and you can write This is a product of intervals. So this is equal to integral security. To our swear to br I'm Santa go from 0 to 2 pi data in the anti derivatives this is one third are cute. R equals 02 times data from 0 to 2 pi evaluating yes, eight thirds times two pi which simplifies to 16 3rd pot

So you want to go ahead and look at this triple integral. The first bound from negative pirate pirate too. The second bound from 0 to 2. The third bound from 02 R. Squared of are busy Trt data now um you can go ahead and since we have prior to and native Power two and data and nothing in this manner. Set integral. Have integral uh involves data. We can go ahead and separate it out and then just evaluate the inner set. So yeah uh you are okay now what does this represent for one? We can go ahead and write our bounds out separately. So we have see is between R squared and zero Bars between two and 0 and in theaters between Higher materials and negative pi over two. So if we go ahead and draw our three space right here we know that the vehicles are squareness. Our upper bound which is a parable Lloyd. Go ahead and brown and blue. Like that's so it's just this little tabloid that encompasses all of um. Okay yeah all through our and so that is our upper bound and see so what our area will end up the volume where volume one of being is the below this. We have everything below 0- two. And are means that we go from max radius of two in the xy plane. And then negative power to a power or two um represents the angle in the positive X. Axis. So it's we go ahead and draw this is X. Y. Z. Are our region? Is this area this red area underneath the paraiba Lloyd? R squared. It's easy but R squared um within our are between zero and two and in the positive X. Direction. Okay so we can go ahead and evaluate this. So our first power negative power to the power to the data just gives us pie. And then we can go ahead and evaluate the inner integral to give us our times R Z squared over two. I'm sorry. Just see. Yeah yeah D R. And then we evaluate Z from $0 square. Yeah. And so we have negative positive pi. Let's go from 0 to of our cube. Pr Okay. Going ahead and plugging in that we can go ahead and integrate again our to fourth of the four premier to to leaving us with To to 4th over four pi Which is just 16/4 or for high as the final answer to this integral.

Were given integral and were asked to sketch solid whose volume is given by this integral and to evaluate the integral. This is the triple integral, which is the integral completed equals negative. However, to be positive, tired too integral from r equals zero r equals two into often t equals zero t equals Where of our T v r data? Well, the region of integration is given in cylindrical coordinates and this is the reading e, which is the set of triples our data Z Such that data lies between negative piles of two and positive pi over two our lives between 02 and Z light between zero and X square, which is the same as R squared. Mhm. Now this solid set represents the solid region above quadrants one and four in the X Y plane enclosed by circular cylinder R equals two and this is bounded above. But the circular proud Lloyd Z equals R squared. This is the same a CD four X squared plus y squared and finally his founded below by the plane. Zico hero. So the X y plane. But it's information you can sketch graph. So we have the X y and Z axis and a draw These in a different configuration than usual, emphasize our region. Now the positive X axis is to the left and the positive y axis system front. And we have a cylinder of radius too. And this is going to intersect. Crab avoid Z equals r squared in the plane C equals four. Somewhere up here, really interested in the 1st and 4th province. Inside this we have crab Lloyd equals R squared, which has a vertex of the origin your sex cylinder in a circle. So look something like this. So this is a drawing of our solid. Now, to find the volume of this solid, this is the integral from failing post negative pi over two positive pi over two. Integral from r equals 0 to 2. Integral from 00 u r squared of our the CDR data. We can take anti derivatives with respect to see you get in the world from negative to positive prior to the girl from 0 to 2. Uh, RG from Z equals zero to z equals r squared He already data evaluating. We have the girl from negative pi over two pi over two and they go from 0 to 2. Um, are you? He argues data intravenous. Their movement. This is a private intervals. The girl from negative to positive pirate too. Data attempts into a 02 are cute D r taking anti derivatives. You get data Navy pilot to pilot the tube times 1/4 are the fourth from 0 to 2. Evaluating this is high times four and it's zero, which simplifies to four pi.

Okay, So we're gonna go ahead and do now is, um show with solid This represents and evaluate this interval. So we have the order of integration, dear. Oh, detained 85. So we know that. Ah, we're bounds are described described Row here data here and five and notice how all of these are constants. So, um, our sops or surface are solid. Actually, they're just going to be ah, section of a sphere. So we have, um, ro is between zero and three. They does between your empire to and buys between to your own pirate six. So if we go ahead and draw clean and this is positive, wide positive X deposits E we know that our row is gonna be describing the sort of this fear. Um, any point from a distance? Three or less. Um, our data describes the first question in the X Y plane and are by describes angle of pi over six between the Z axis. So we're gonna end up with this sort of section of a cone? Not really. Account a sphere spherical section stemming from center. So we have a sphere. We have, like this spherical. Um, I want to say sector because that's the closest thing I can think of when it comes to like a circle, like a sector a circle. But it's a spherical sector, and now we wouldn't, um, go ahead and evaluate this. So we have Maria General, it's pretty simple to evaluate. Um, so they do, uh, defy. And, um, we'll go ahead and evaluate this so we have Ah, dear. Oh, first. So we only worry about our road terms. So a row is row squared and in a row, row squared is row Q divided by three. And, um three. Cubed over three is nine and zero Q zero. So we end up with nine and I'll bring it out front. And so we're left with the outer into girls signed by, Do you? They identify. And since there's no data terms, we could just pull out a pirate too, and evaluate our final winner, girl Ah, which just becomes native co sign at night of Coastline Fi from zero to Pirates six and we know that this gives us co sign zero minus co sign pi over six. So our final answer is just nine pi over two times one minus square three on two. So this is our total volume. Um, because we know that, um, this is just are are in a grand is just be a written in spherical coordinates. So this is the volume nine, Tyra, two times one minus screwed over to scroll through over two. And it describes the area of this section of the sphere that, um, is described by these constant these ranges of constance from Rhode Data and by


Similar Solved Questions

5 answers
26_ Solve the following differential equations "-2y+6y=0 b) y+2y'-8y=0 25y" 20y'+4y=0 -3y"+3y'-y=0
26_ Solve the following differential equations "-2y+6y=0 b) y+2y'-8y=0 25y" 20y'+4y=0 -3y"+3y'-y=0...
5 answers
Calculate the concentration (in molecules /cm? and the mixing ratio (in ppIn) of water vapour at ground level at T = 298 K at RH values of50% 70% 90% 99%
Calculate the concentration (in molecules /cm? and the mixing ratio (in ppIn) of water vapour at ground level at T = 298 K at RH values of 50% 70% 90% 99%...
5 answers
Evaluate the integral by changing the order of integration in an appropriate waySSf 9xz ezy dy dx dzSS f Ixz ezy2 dy dx dz = (Type an exact answer )
Evaluate the integral by changing the order of integration in an appropriate way SSf 9xz ezy dy dx dz SS f Ixz ezy2 dy dx dz = (Type an exact answer )...
5 answers
Flx) = Flnd the -/2 points exponentlal Se55 functlon . 2 Ca whose graph 3-12 points Submit Answer Need Help? Praclice Rooa h SEssCalcET? 3 1 Ena Another Version 1
flx) = Flnd the -/2 points exponentlal Se55 functlon . 2 Ca whose graph 3 -12 points Submit Answer Need Help? Praclice Rooa h SEssCalcET? 3 1 Ena Another Version 1...
5 answers
'fe: drdyover the region that bounded by the curve y vand the line = 24 Evaluate the integralf _Select one: 45-16{-26;-} 4{-1
'fe: drdyover the region that bounded by the curve y vand the line = 24 Evaluate the integralf _ Select one: 45-1 6{-2 6;-} 4{-1...
5 answers
The mass ol Ihe planet Jupiter MJ=1.97 x 1027 kg, ard its (udL surface of Jupiter?7.15 10/ m. Whalacceleralion ol gravity on Ihe25.7 rs28.87 m/sz1.60 m/s210.A4 m/s23.72 m/s2
The mass ol Ihe planet Jupiter MJ=1.97 x 1027 kg, ard its (udL surface of Jupiter? 7.15 10/ m. Whal acceleralion ol gravity on Ihe 25.7 rs2 8.87 m/sz 1.60 m/s2 10.A4 m/s2 3.72 m/s2...
5 answers
What is the configuration of the stereogenic center in the compound below?NH2Me
What is the configuration of the stereogenic center in the compound below? NH2 Me...
5 answers
Why is 1 not allowed as a base for a logarithmic function?
Why is 1 not allowed as a base for a logarithmic function?...
5 answers
In the rock-paper-scissors experiment, if B you win, how many outcomes are in BC ?
In the rock-paper-scissors experiment, if B you win, how many outcomes are in BC ?...
1 answers
Use the Change of Base Formula to show that $$\log e=\frac{1}{\ln 10}$$
Use the Change of Base Formula to show that $$\log e=\frac{1}{\ln 10}$$...
5 answers
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soot to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)What is the percent chance that all the children attending the party prefer soda?
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soot to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.) What is the percent chanc...
5 answers
(Order Notation) . Let f(x) and g(z) be functions of x taking values that are positive We say f is big-0 of g" and write f() = O(g(x)) if there are positive constants and C such that f() < c g(r) Vx > C.
(Order Notation) . Let f(x) and g(z) be functions of x taking values that are positive We say f is big-0 of g" and write f() = O(g(x)) if there are positive constants and C such that f() < c g(r) Vx > C....
4 answers
8. Simplify: Write your answers with positive [email protected] 0lb
8. Simplify: Write your answers with positive exponents. @b 0lb...
5 answers
Answer answer box within your l '94 Predict the original original data. It is not meaningful t0 dala; girth data It is not meaningful t original data box wilhin your not meaningful girth meaningful 1 cheoigth Ij pipaud Vi 164 predict 1 VL 5141 172 Amea 1 Om 3 IU meaninglul H because x = meaningful 164 needed ) Select the 172 1 1 inside 1 V 1 cnoice 1 1 correct choic in tne Tanye Ole below 1 1 belaw and; 1 necesSurY,
answer answer box within your l '94 Predict the original original data. It is not meaningful t0 dala; girth data It is not meaningful t original data box wilhin your not meaningful girth meaningful 1 cheoigth Ij pipaud Vi 164 predict 1 VL 5141 172 Amea 1 Om 3 IU meaninglul H because x = meaning...
5 answers
#1) Find paramettic exquations for tlu tAngent line to the curve with th' giveu parametric" equations nt the specified pint .lug ((0,45)
#1) Find paramettic exquations for tlu tAngent line to the curve with th' giveu parametric" equations nt the specified pint . lug ( (0,45)...

-- 0.021422--