Calc: Find a definite integral for the mass

[calculusisfun]

Homework Statement

A cylinder mug with a 4 centimeter radius and 8 centimeter height is filled with tea. When sugar is added to the tea, the sugar usually settles to the bottom of the cylinder mug. The density of sugar in the tea at a height h centimeter from the bottom of the cylinder mug is given by the following formula: d(h) = 0.01(8-h). [Note: mass = (volume)(density)]

Write a definite integral that gives the mass of sugar in the mug exactly.

2. The attempt at a solution

Volume of the mug: [PLAIN]http://img839.imageshack.us/img839/2599/calc3.png

Mass = volume x density

Mass = [PLAIN]http://img215.imageshack.us/img215/3589/calc4.png

Not sure if I did this right or completely wrong.

Help would be much appreciated. :)

 

[tiny-tim]

hi calculusisfun !

no, the cup is divided into horizontal slices of height dh,

so inside your volume-times-density integral, you must put the volume of the slice …

it'll be a mutliple of dh

(that's where your "dh" comes from! )

(you've used the volume of the whole cup, which doesn't give the same result)

 

[calculusisfun]

Thanks for responding tiny-tim! :)

So, would it be like this? [PLAIN]http://img80.imageshack.us/img80/9790/calc5.png

I multiplied d(h) by 16 pi h, because 16 pi h represents the height of an individual slice of the cylinder.

Am I still doing this completely wrong? :p

I feel like such a nerd staying in on Halloween haha, but any help is much appreciated. :)

 

[calculusisfun]

Can anyone verify? Apologies for the bump.

 

[tiny-tim]

hi calculusisfun!

(just got up :zzz: …)

I multiplied d(h) by 16 pi h, because 16 pi h represents the height of an individual slice of the cylinder.

(have a pi: π )

no, you haven't quite got the hang of this slicing business

the slice is only dh high

(with these slices, the d-something is always one of the dimensions, in this case dh is a length, so you only need the area, you don't need an extra h )

since its area is πr², that's a total volume of πr² dh …

in this case, r is constant, so it's only 16π dh (not 16π h dh)

(btw, i'd keep the 16π*.01 outside a bracket, until just before the end)

try again! ​

 

[calculusisfun]

Thanks for the patience tiny-tim. :)

So, are you saying it should actually be:

[PLAIN]http://img820.imageshack.us/img820/3233/eqn3042.png

??

I'm confused. If mass = (density)(volume), why can't I multiply those two formulas together and then integrate??

Sorry for being so uneducated. :p

 

[tiny-tim]

hi calculusisfun!

(just got up :zzz: …)

I'm confused. If mass = (density)(volume), why can't I multiply those two formulas together and then integrate??

you have done …

0.01(8 - h) is the density, and 16πdh is the volume (of each slice)

you need to convince yourself of this!

go through some worked examples from your book (i expect they have things like volume of a sphere from slices … that's with constant density of course) to see it in operation

(and of course come back here if you're still worried )

 

[calculusisfun]

Hey Tin-tim,

Again thanks for your help. :)

This question was just on a handout my teacher gave me, so I went looking online for similar problems. I came across this one here.

It looks like they did what I suppose I did. They multiplied the volume function by the density function and then integrated.

It just seems to make sense to me this way haha.

v(h) = 16πh <---- volume of cylinder h high
d(h) = 0.01(8-h) <----- density of cylinder h from the base

So then just multiply them together and integrate that formula from 0 to 8. Why is this incorrect? You said it adds an extra h or something along those lines, but the h in the volume formula and the h in the density formula are independent of each other it appears.

Haha geez I am confused. :p

 

[tiny-tim]

hey calculusisfun!

… I came across this one here.

It looks like they did what I suppose I did. They multiplied the volume function by the density function and then integrated.

i'm having difficulty viewing that link, but i think there's no integration involved

It just seems to make sense to me this way haha.

v(h) = 16πh <---- volume of cylinder h high
d(h) = 0.01(8-h) <----- density of cylinder h from the base

So then just multiply them together and integrate that formula from 0 to 8. Why is this incorrect? You said it adds an extra h or something along those lines, but the h in the volume formula and the h in the density formula are independent of each other it appears.

ah, when i said "you don't need an extra h", i was talking about dimenions

if you write the area as πr² (instead of 16π), it becomes clearer …

it should be ∫ (density) πr² dh, which is ∫ (density) (a volume),

but you're trying ∫ (density) πr² h dh, which is ∫ (density) (a volume) (a tiny distance)!

tiny distances are still distances!

to integrate the density times the volume, essentially you do ∫ (density) d(volume), and in this case d(volume) is the volume of the tiny slice

 

[calculusisfun]

Thanks for the prompt reply! :)

Oh, so you're saying the volume of a slice is πr^2? But don't you have to take account of the height of the slice? So you would need the h right?

 

[tiny-tim]

Thanks for the prompt reply! :)

Oh, so you're saying the volume of a slice is πr^2? But don't you have to take account of the height of the slice? So you would need the h right?

no, the volume of a slice is πr² dh

dh is a length

(and it's not h because you only want the slice )

(btw, I'm going out for the evening by half-past)