Question on exponential intergration (1 Viewer)

[squeenie]

This question is from Maths in Focus (revised edition), page 152

8. Use Simpson's rule with 5 function values to find the volume of the solid formed when the curve y=e^x is rotated about the y-axis from y=3 to y=5, correct to 2 significant figures.

According to the book, the answer is 12, but every time I've tried it, I always seem to get around 8.64, depending on the method I use (I've done Simpson's rule as well as regular intergration and graphing the equation)

Can someone tell me if I'm right?

 

[tommykins]

回复: Question on exponential intergration

i've never used the simpsons rule to find the volume of a solid, but i'm guessing you need to apply it to x^2 dy and then multiply that by pi.

y = e^x -> x = ln y -> x^2 = (lny)^2

v = pi * simpsons rule + (lny)^2

I'd have no clue how you'd integrate lny with only 2unit knowledge.

Shit question IMO

 

[squeenie]

Re: 回复: Question on exponential intergration

Shit question IMO

You know, that's what I was thinking too. I tried re-arranging to lny^2 using log laws, but that just made things worse. Oh well. Maths in Focus isn't a really good textbook anyways.

 

[me121]

the method tommykins suggested is proably what they expect..

but an alternate method (which would give different results) would be to take an even grid of points under the surface and fit paraboloids to each 2D section.. then sum these up..


but to int ln(y).. we can just integrate exponential function and use the fact that the inverse of ln is exp

 

[squeenie]

but an alternate method (which would give different results) would be to take an even grid of points under the surface and fit paraboloids to each 2D section.. then sum these up..

So would that be something like the trapezoidal rule?

 

[me121]

So would that be something like the trapezoidal rule?

no.

the method that they probably expect (and is probably correct, and is what i did when i was doing the hsc) would be to use Pi*int(x^2) dy (or y^2 depending on which axis), but use simpsons rule to evaluate the integral.

the above method is what you should do.

i was going to far in mentioning that other method, and you would never do it in an exam. it is a whole different approach to the problem..

if you really are interested, then continue on.
================================= (this should really go in the extra-curricula maths sub-forum..)
here is what i was thinking.. ...an interesting way of looking at the problem is to use an extended version of simpsons rule for double integrals (i.e. area under a surface, i.e. to find volumes)..

the trapezoidal rule joins a straight line between points and find simple areas.. simpsons rule fits parabolas in between points and uses the fact that we know how to evaluate the area under a parabola.

why not extend this to splitting the surface into a square grid and fit paraboloids to the 4 grid points. (not sure if this is possible).. then use the fact that you know (or at least can try to work out) the volume under a paraboloid.

i'm interested to see what Derek has to say about this. because i don't know enough maths to know what i am talking about.

 

[Mark576]

This question is from Maths in Focus (revised edition), page 152

8. Use Simpson's rule with 5 function values to find the volume of the solid formed when the curve y=e^x is rotated about the y-axis from y=3 to y=5, correct to 2 significant figures.

According to the book, the answer is 12, but every time I've tried it, I always seem to get around 8.64, depending on the method I use (I've done Simpson's rule as well as regular intergration and graphing the equation)

Can someone tell me if I'm right?

y=e^x
x=lny => x²=(lny)²
V = pi[3-->5]∫(lny)²dy
Using simpson's rule, we first evaluate [3-->5]∫(lny)²dy:

Hence:
I=2/12[(ln3)²+(ln5)²+4((ln3.5)²+(ln4.5)²)+2(ln4)²]=3.827919825

Now to find the volume, we need to multiply by pi which gives: 12.0257648

 

[squeenie]

y=e^x
x=lny => x²=(lny)²
V = pi[3-->5]∫(lny)²dy
Using simpson's rule, we first evaluate [3-->5]∫(lny)²dy:

Hence:
I=2/12[(ln3)²+(ln5)²+4((ln3.5)²+(ln4.5)²)+2(ln4)²]=3.827919825

Now to find the volume, we need to multiply by pi which gives: 12.0257648

Hey, thanks! I tried that before, but I got something else other than 2/12 at the start, somehow.

 

[tommykins]

y=e^x
x=lny => x²=(lny)²
V = pi[3-->5]∫(lny)²dy
Using simpson's rule, we first evaluate [3-->5]∫(lny)²dy:

Hence:
I=2/12[(ln3)²+(ln5)²+4((ln3.5)²+(ln4.5)²)+2(ln4)²]=3.827919825

Now to find the volume, we need to multiply by pi which gives: 12.0257648

Thanks for filling in my laziness

 

[conics2008]

Hey in 2unit if you cant integrate a function with 2unit knowledge always use approx areas and then to find volume multiply it with pi =)

thats the 2unit stuff =)

or you can use this rule... integration by parts...

use the fact that S ln(x) = x(ln(x)-1)

therefore by parts

ln(x)^2 = ln(x).ln(x)

therefore int = x[ln(x)]^2 - 2xln[x] + 2x sub in your limits xD

good day =)