Register to reply

Integration question again

by lionely
Tags: integration
Share this thread:
lionely
#1
Feb23-13, 11:02 AM
P: 523
1. The problem statement, all variables and given/known data
A goldfish bowl is a glass sphere of inside diameter 20cm. Calculate the volume of water it contains when the maximum depth is 18cm.





The attempt at a solution

I don't really have an idea of how to attempt this, all I did so far was a draw a little sketch of the bowl and put in the dimensions

Hmm.. should I just find the volume of the sphere, sketch the cross-section of it, and then try to use the principles of solids of revolution to find the volume?
Phys.Org News Partner Science news on Phys.org
Sapphire talk enlivens guesswork over iPhone 6
Geneticists offer clues to better rice, tomato crops
UConn makes 3-D copies of antique instrument parts
Curious3141
#2
Feb23-13, 11:26 AM
HW Helper
Curious3141's Avatar
P: 2,943
Quote Quote by lionely View Post
1. The problem statement, all variables and given/known data
A goldfish bowl is a glass sphere of inside diameter 20cm. Calculate the volume of water it contains when the maximum depth is 18cm.





The attempt at a solution

I don't really have an idea of how to attempt this, all I did so far was a draw a little sketch of the bowl and put in the dimensions

Hmm.. should I just find the volume of the sphere, sketch the cross-section of it, and then try to use the principles of solids of revolution to find the volume?
Consider the vertical cross-section of the bowl. Let the centre of the bowl be the origin (0,0). y-coordinates range from -10 to +8.

Now consider the horizontal circular cross section of a disc of water taken at a certain y-coordinate. Find its radius via Pythagoras theorem, and hence its area.

Hence figure out the volume of an infinitesimally small cylinder having that cross-section and a vertical height dy.

Now do the integration, imposing the correct bounds for y.
lionely
#3
Feb23-13, 11:39 AM
P: 523
Umm wouldn't the radius be 10cm, because the diameter of the bowl is 20cm?

Cause maybe I drew my diagram badly I'm not seeing how I can use Pythagoras ' theorem.
:S

Curious3141
#4
Feb24-13, 12:00 AM
HW Helper
Curious3141's Avatar
P: 2,943
Integration question again

Quote Quote by lionely View Post
Umm wouldn't the radius be 10cm, because the diameter of the bowl is 20cm?

Cause maybe I drew my diagram badly I'm not seeing how I can use Pythagoras ' theorem.
:S
Right through the centre (at y = 0), yes the radius of the cross-section would be 10cm.

But what about other y values. Hint: think of a right triangle, the hypotenuse being the constant radius of the bowl (10cm), the vertical height being y and the horizontal base being the radius of the cross-section.
lionely
#5
Feb24-13, 11:23 AM
P: 523
Is the radius 6cm?
Curious3141
#6
Feb25-13, 12:40 AM
HW Helper
Curious3141's Avatar
P: 2,943
Quote Quote by lionely View Post
Is the radius 6cm?
Huh? The radius of the horizontal cross-section varies continuously depending on the level you're taking it at.

Did you make a proper sketch?
lionely
#7
Feb25-13, 06:22 PM
P: 523
Hm... Maybe I didn't on my diagram 10 is the hypotenuse and 8 is the perpendicular height.
Curious3141
#8
Feb26-13, 03:48 AM
HW Helper
Curious3141's Avatar
P: 2,943
Quote Quote by lionely View Post
Hm... Maybe I didn't on my diagram 10 is the hypotenuse and 8 is the perpendicular height.
10cm is always the hypotenuse.

y varies from -10 (bottom) to +8 (top of water level). Do you understand this?

8cm is only the height when you're considering the area of the top surface of the water (y = 8). At that point, the radius is ##\sqrt{10^2 - 8^2} = 6cm##. Agree?

At the bottom of the bowl (y = -10), the radius is zero, because the bottom is just a point, not a circle. Agree?

The radius of the cross-section right through the level of y = 0 (center of the sphere) is 10cm (simply the radius of the sphere). Agree?

Now you can take the cross section of water at *any* water level between the bottom and the top, not just those "special" levels. Your job is to find an expression, in terms of y, for the radius of this cross-section. Can you do this?

Remember, what you get will be in terms of y - it'll have a y in the expression, not just a number.


Register to reply

Related Discussions
Integration question Calculus 0
Integration question Calculus 1
Integration question Calculus 4
Question arrising from integration homework (advanced integration i guess?) Calculus 9
Integration Question General Math 3