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Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a ...

Question

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume $a, b, c, r, R,$ and $h$ are positive constants.Spherical cap Find the volume of the cap of a sphere of radius $R$ with thickness $h$.(FIGURE CAN'T COPY)

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume $a, b, c, r, R,$ and $h$ are positive constants.Spherical cap Find the volume of the cap of a sphere of radius $R$ with thickness $h$. (FIGURE CAN'T COPY)



Answers

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume $a, b, c, r, R,$ and $h$ are positive constants.Spherical cap Find the volume of the cap of a sphere of radius $R$ with thickness $h$. (FIGURE CAN'T COPY)

In this question be equals trouble integration. All Bruce quit. Signed by do you know defy the Seeker. Find the tradition limits. Zoo for Tzeitel boy were city Tzeitel Boy, If we think the first integration, we will have 64/3 Signed Boy de Foix, the CEO. If you think another one, we will have 32 adversity integration off the Sita which is equal 64 boy over sleep.

In this question be Is the whole travel integration off through square sign boy dear Oh, you boy D C on the integration limits. Zero to body over six by over city fetal for you were, too. While taking the integration, we can get 8/3. Fine boy. Defy these either, Which is equal for over city Rude City minus one. Integration off the seat, which is equal to buy over three Rude city minus one

So we're going to be using a triple integral to find the volume of this given shape. So we have the sphere X squared plus y squared. Plus, these great was 19 as this top curve and Z squared minus X squared minus y squared equals one as this lower curve. So we're going to start out finding thes e boundaries for this so that for the upper boundary will have subtracting X squared and Y squared from both sides of 19 minus X squared minus y squared equals C squared. Square root of both sides is positive or negative. Square root of 19 minus X squared minus y squared equals Z. We're only concerned about positive numbers for Z, though, so we're just going to have our upper limit. Frizzy Z is less than or equal to square root of 19 minus X squared minus y squared. Next, we're going to be looking at this, the lower bound for Z, so that's going to be if we add X squared y scored both sides. The squared equals one plus X squared, plus y squared square root of both sides of C equals positive or negative square root of one plus X squared plus y squared again, we're only concerned about positive values. Frizzy. So our lover limit is going to be square root of one plus X squared. Plus y squared is less than or equal to see. So now we're going to look at what ex and why are equal to. So we're going to be looking at where X and wire whitest because that's where and that's going to be where these two curves meat. If you can see on this graph, so we're going to have when square root of one plus X squared plus y squared equals square root of 19 minus X squared minus y squared is going to be squaring both sides. One plus X squared plus y squared equals 19 minus X squared minus y squared, adding X squared and Y squared to both sides and subtracting one. We get two X squared plus two y squared equals 18 fighting both sides by two. So we'll get X squared plus y squared equals nine. So that means that we have a circle with a radius of nine in our X Y plane. There, a circle with a radius of three in our X Y plane. So to set up this integral, it might be easier if we use cylindrical coordinates. So those are going to be going to have a triple triple integral of our the radius. The DZ de are de fada. So we already have our bounds for Z. So if x squared plus y squared equals R squared that's defined for us in the textbook and in multiple places. So X squared plus y squared equals R Square. So every time we see X squared plus y squared, we can substitute that. So we'll have our lower bound for Z being one plus r squared and our upper bound being square root of 19 minus r squared for Are we already said that our that we have a radius of three. So we're going to have 0 to 3 for a radius and finally fade. It goes all the way around this circle, so we're going to go from 0 to 2 pi. So we go in a full circle. So now we have this integral set up so we can actually evaluate this integral. I'm going to scroll down so that we can focus just on evaluating this integral. So we're going to leave the data and are into girls on the outside. Take the anti derivative of our with respect disease of our times e evaluated from square root of one plus R squared to square root of 19 minus R squared and later we're going to be evaluating with respect to the radius and data. So we're going to plug in the upper and lower balance for Aziz will have our times square of 19 minus R squared minus R times square, root of one plus R squared with respect to our data. So using the fundamental theory of calculus, we know that we can separate these two into to separate into girls. So I'm going to separate these so that it might be a little easier for us to evaluate them. So these are going to be to separate into girls here still, with respect to our in data. So now we're going to use U substitution for this 1st 1 So I'm going to set you equal to 19 minus r squared this ah, term that's inside of the square room. So the derivative with respect to our is equal to negative two times are. So we're gonna have negative 1/2. Do you equal to r D R So we can use substitution now, so we'll have the integral from 0 to 2 pi. The inner girls of RDR is going to be replaced with negative 1/2 do you? And square root of Nick 19 minus R squared is gonna be replaced by square root of you. So we're gonna have this. We need to find our lower and upper bounds, though. So when R equals zero U equals 19 minus zero squared, so you equals 19. And for our upper bound u equals 19 minus three squared Sonae U equals 10. So that's going to be our upper bound there. So now we're going to do the same thing for our next under for an extended girl. So we're going to have do you get in a girl from 0 to 2 pi thean a girl from we don't know these bounds yet, so we're going to set you equal to one plus r squared the derivative with respect to our is going to be to art. So we're gonna have 1/2 times do you equal to RDR. So we're replacing RTR with 1/2 to you. So 1/2 do you square root of you and now we need to find the bounds. So when R equals zero U equals one plus zero, so U equals one. And for the upper bound U equals one plus three squared nine. So you equals 10. And don't forget that we're also evaluating this with respect of data. So I'm going to this girl this down so that we can evaluate this integral. So we're going to go from 0 to 2 pi going to take the anti derivative of negative 1/2 times you to the 1/2. So thank you. 5 1/2 times you raise that power to buy once of three have times the reciprocal of that new exponents evaluated from 19 to 10 minus being a girl from 0 to 2 pi of 12 1 1/2 times you to the three house raising that exponents times 2/3 evaluated from 1 to 10. So now we're going to plug in 10 and 19 So we'll have negative 1/2. Well, these two and these twos canceled out. So negative 1/3 times 10 to the three halves minus negative. 1/3 time's 19 to the three house minus one have minus the integral from 0 to 2 pi. That's right. In this deflated right there, Man is integral from 0 to 2. Pi of 1/2 were 1/3 thes two zehr, cancelling out so 1/3 times 10 to the three halves minus 1/3 times, one to the three house with respect of data. So since these are both being evaluated from 0 to 2 pi with respect to theta weaken now combine them into one integral. So have the integral from 0 to 2 pi of negative 10 square area to 10 is the is 10 to the three have over three plus 19 square root in 19/3, minus 10 square root to 10 over three, plus 1/3. So we can all we can put these all into one fraction now so and combine like terms. So we'll have one my plus 19 square root in 19 minus 20. Scary to 10 over three with respect to fail us. And now we can evaluate this with respective beta so we'll have. This is all one a constant one plus 19 square root of 19 minus 20 square to 10 over three, multiplied by data evaluated from 0 to 2 pi. So if we multiply this by two, pi will get two pi times one plus 19 square root of 19 minus 20 square root of 10 over three minus zero. So this is going to be the volume for this shape.

So we're going to be using a triple integral to find the volume of this given shape. So we have the sphere X squared plus y squared. Plus, these great was 19 as this top curve and Z squared minus X squared minus y squared equals one as this lower curve. So we're going to start out finding thes e boundaries for this so that for the upper boundary will have subtracting X squared and Y squared from both sides of 19 minus X squared minus y squared equals C squared. Square root of both sides is positive or negative. Square root of 19 minus X squared minus y squared equals Z. We're only concerned about positive numbers for Z, though, so we're just going to have our upper limit. Frizzy Z is less than or equal to square root of 19 minus X squared minus y squared. Next, we're going to be looking at this, the lower bound for Z, so that's going to be if we add X squared y scored both sides. The squared equals one plus X squared, plus y squared square root of both sides of C equals positive or negative square root of one plus X squared plus y squared again, we're only concerned about positive values. Frizzy. So our lover limit is going to be square root of one plus X squared. Plus y squared is less than or equal to see. So now we're going to look at what ex and why are equal to. So we're going to be looking at where X and wire whitest because that's where and that's going to be where these two curves meat. If you can see on this graph, so we're going to have when square root of one plus X squared plus y squared equals square root of 19 minus X squared minus y squared is going to be squaring both sides. One plus X squared plus y squared equals 19 minus X squared minus y squared, adding X squared and Y squared to both sides and subtracting one. We get two X squared plus two y squared equals 18 fighting both sides by two. So we'll get X squared plus y squared equals nine. So that means that we have a circle with a radius of nine in our X Y plane. There, a circle with a radius of three in our X Y plane. So to set up this integral, it might be easier if we use cylindrical coordinates. So those are going to be going to have a triple triple integral of our the radius. The DZ de are de fada. So we already have our bounds for Z. So if x squared plus y squared equals R squared that's defined for us in the textbook and in multiple places. So X squared plus y squared equals R Square. So every time we see X squared plus y squared, we can substitute that. So we'll have our lower bound for Z being one plus r squared and our upper bound being square root of 19 minus r squared for Are we already said that our that we have a radius of three. So we're going to have 0 to 3 for a radius and finally fade. It goes all the way around this circle, so we're going to go from 0 to 2 pi. So we go in a full circle. So now we have this integral set up so we can actually evaluate this integral. I'm going to scroll down so that we can focus just on evaluating this integral. So we're going to leave the data and are into girls on the outside. Take the anti derivative of our with respect disease of our times e evaluated from square root of one plus R squared to square root of 19 minus R squared and later we're going to be evaluating with respect to the radius and data. So we're going to plug in the upper and lower balance for Aziz will have our times square of 19 minus R squared minus R times square, root of one plus R squared with respect to our data. So using the fundamental theory of calculus, we know that we can separate these two into to separate into girls. So I'm going to separate these so that it might be a little easier for us to evaluate them. So these are going to be to separate into girls here still, with respect to our in data. So now we're going to use U substitution for this 1st 1 So I'm going to set you equal to 19 minus r squared this ah, term that's inside of the square room. So the derivative with respect to our is equal to negative two times are. So we're gonna have negative 1/2. Do you equal to r D R So we can use substitution now, so we'll have the integral from 0 to 2 pi. The inner girls of RDR is going to be replaced with negative 1/2 do you? And square root of Nick 19 minus R squared is gonna be replaced by square root of you. So we're gonna have this. We need to find our lower and upper bounds, though. So when R equals zero U equals 19 minus zero squared, so you equals 19. And for our upper bound u equals 19 minus three squared Sonae U equals 10. So that's going to be our upper bound there. So now we're going to do the same thing for our next under for an extended girl. So we're going to have do you get in a girl from 0 to 2 pi thean a girl from we don't know these bounds yet, so we're going to set you equal to one plus r squared the derivative with respect to our is going to be to art. So we're gonna have 1/2 times do you equal to RDR. So we're replacing RTR with 1/2 to you. So 1/2 do you square root of you and now we need to find the bounds. So when R equals zero U equals one plus zero, so U equals one. And for the upper bound U equals one plus three squared nine. So you equals 10. And don't forget that we're also evaluating this with respect of data. So I'm going to this girl this down so that we can evaluate this integral. So we're going to go from 0 to 2 pi going to take the anti derivative of negative 1/2 times you to the 1/2. So thank you. 5 1/2 times you raise that power to buy once of three have times the reciprocal of that new exponents evaluated from 19 to 10 minus being a girl from 0 to 2 pi of 12 1 1/2 times you to the three house raising that exponents times 2/3 evaluated from 1 to 10. So now we're going to plug in 10 and 19 So we'll have negative 1/2. Well, these two and these twos canceled out. So negative 1/3 times 10 to the three halves minus negative. 1/3 time's 19 to the three house minus one have minus the integral from 0 to 2 pi. That's right. In this deflated right there, Man is integral from 0 to 2. Pi of 1/2 were 1/3 thes two zehr, cancelling out so 1/3 times 10 to the three halves minus 1/3 times, one to the three house with respect of data. So since these are both being evaluated from 0 to 2 pi with respect to theta weaken now combine them into one integral. So have the integral from 0 to 2 pi of negative 10 square area to 10 is the is 10 to the three have over three plus 19 square root in 19/3, minus 10 square root to 10 over three, plus 1/3. So we can all we can put these all into one fraction now so and combine like terms. So we'll have one my plus 19 square root in 19 minus 20. Scary to 10 over three with respect to fail us. And now we can evaluate this with respective beta so we'll have. This is all one a constant one plus 19 square root of 19 minus 20 square to 10 over three, multiplied by data evaluated from 0 to 2 pi. So if we multiply this by two, pi will get two pi times one plus 19 square root of 19 minus 20 square root of 10 over three minus zero. So this is going to be the volume for this shape.


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