How do I find the volume of this?

[es91]

Homework Statement

Find the volume of the object generated when the graph in quadrant 1 from y=0 to y=10 is spun around the y axis.

Homework Equations

http://img.skitch.com/20100102-1qahghkqtbny3c1yyurmqdtr42.png

The Attempt at a Solution

I have no idea how to solve for x in terms of y to integrate the function.

 

[andylu224]

Use the Shell Method instead.

 

[es91]

Ok, so if i want to find the volume generated by revolving the figure b/w the function and the y-axis around the y-axis from y=0 to y=9.8, y(4.7)=9.2 ...Is this correct?

[(9.2)(π)(4.7)^2] - [2π*int(f(x),x,1,4.7)]

Edit: Also, if I want to find the surface area of this solid, can I use the formula 2π*int(x*ds,x,1,4.7) where ds=sqrt(1+f'(x)^2) ?

and how do I find the work required to drain water out of this solid and the rate at which the water level is rising when the solid is 1/2 full if h2o is being poured at a constant rate of 2 cm^3/s

Thank you.

 

[es91]

I am still stuck on this one, help greatly appreciated.

 

[yungman]

This is my approach, take it as a suggestion:

Revolving around y-axis means the solid generated made up of a stack of round disc with $$r=\sqrt{x^{2}+z^{2}}$$ with thichness of dy.

Find the equation of r, then use r to get the area of the disc and integrate resp to y from y=1 to 10 to get the volume.

 

[es91]

Isn't it impossible to find r?

 

[LCKurtz]

Ok, so if i want to find the volume generated by revolving the figure b/w the function and the y-axis around the y-axis from y=0 to y=9.8, y(4.7)=9.2 ...

Why are you using 9.8 when the problem says y = 10?

[(9.2)(π)(4.7)^2] - [2π*int(f(x),x,1,4.7)]

What does y = 9.2 and x = 4.7 have to do with this problem?

 

[yungman]

Isn't it impossible to find r?

Sorry I did not see the equation on the top of the graph.

From the equation, the graph is on xy plane only, so r=x. From that you generate the equation of the area and integrate resp to y to get the volume.

For surface area use r to get the circumference, integrate resp y to get surface area.

 

[es91]

Ok, change the 10 to a 9.2 and y(4.7)=9.2 ...

 

[LCKurtz]

Ok, change the 10 to a 9.2 and y(4.7)=9.2 ...

And why is your lower limit on x = 1? That is not where the curve touches the x axis.

 

[es91]

Sorry, it's 0.5. Is the rest correct and do you know how to do the other parts?

 

[LCKurtz]

Sorry, it's 0.5. Is the rest correct and do you know how to do the other parts?

With the correct limits both your volume and surface area should work. To calculate the work done emptying the vessel, perhaps you could find the center of mass (it's on the axis of revolution) and treat the water like a point mass at that location.

I presume you are using a program like Maple of Mathematica to help with the calculations, right?

 

[es91]

I'm in Calc. BC and have no knowledge of those programs.

 

[LCKurtz]

I'm in Calc. BC and have no knowledge of those programs.

What are you using to solve where y = 0 (of which your answer is only correct to 1 decimal place? And why did you change the problem to y = 9.2 instead of y = 10? Can't you use the same process to solve for x when y = 10 that you used to solve for x when y = 0? You must be using some type of calculating device, no?

 

[es91]

The equation was generated by fitting a curve to a series of points.

 

[LCKurtz]

The equation was generated by fitting a curve to a series of points.

Now you tell us. You know, it would save everyone lots of time if you would explain the whole problem up front. And maybe tell us what you are studying that is relevant. Simpson's rule? Least squares fit? Trapezoidal rule? Polynomial interpolation? You may be asking for help on an approach that isn't even the way to do your problem.

 

[es91]

The curve fitting was not part of the problem; I just needed to find an equation for a 3d object and now I need to solve those problems using the equation.

 

[LCKurtz]

The curve fitting was not part of the problem; I just needed to find an equation for a 3d object and now I need to solve those problems using the equation.

Let's see...you have data points (which you haven't given us). Curve fitting isn't part of the problem but you gave us a curve. You need the equation of a 3d object. Any 3d object?? Does the object have anything to do with the data points? Why not just use a parabola to revolve?

Please give us a complete un-edited statement of the problem you are trying to solve and what you have tried so far. Otherwise I feel like I am wasting my time with this thread.