Hey there! Sign in to join this conversationNew here? Join for free
Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
    hi guys, i've just learnt the shell and disc method for finding the volume using integration. however, i do not know when to use which one. could anyone please give me some tips?
    thanks
    Offline

    2
    ReputationRep:
    In general when a curve is rotated around the x-axis between x = a and x = b the volume is
    \displaystyle \int_a^b{\pi y^2 \mathrm{d}x} = \pi \int_a^b{y^2 \mathrm{d}x}
    In the case of the curve being rotated around the y-axis, you simply swap x and y in there.

    What more do you need to know? :cool:
    Offline

    8
    ReputationRep:
    (Original post by kingsclub)
    hi guys, i've just learnt the shell and disc method for finding the volume using integration. however, i do not know when to use which one. could anyone please give me some tips?
    thanks
    It depends on the axis of revolution (parallel with x or y) and
    on the y=f(x) function revolved.

    With disc method we slice the area of revolution perpendicular to the axis. Let this axis be the x or parallel line with it (equation is a constant y0 value). We caculate an elementary cylindric volume from the disk area and the elementary \Delta x height. We had to write up the radius in function of x. This will be the f(x) itself if the axis is x (y0=0) or f(x)-y0=R(x).
    With limit of Delta to zero \Delta x \rightarrow dx and summing the elementary volumes we get
    V=\pi \cdot \int^b_a (R(x))^2 dx

    If hard to integrate this function and would be more simple to work with the
    inverse function then we can use the shell method or vice versa.

    With shell method we slice the area of revolution parallel with the axis at an y value. The length of the slice will be the height of the cylinder and the y value is the radius. THe elementary volume is from this cylinder surface and a \Delta y pipe thickness. The y value is change between the lower and upper bound.
    From y=f(x) -> x=g(y) which is the inverse, and we can caculate the shell height from the general x
    with g(y) and get it in function of y. Height =g(y2)-g(y) (g(y2)=b as above). With limit of \Delta y \rightarrow dy and summing we get the volumes as integral of
    V=2\pi \int^{y_2}_{y_1} (y-y_0)\cdot (g(y_2)-g(y))\ dy
    where g(y_2) is the given upper x value that is b in the other method, and y_0 is the equation of the horizontal line of revolution, which is a constant value, and 0 if the axis ia the x.

    Both method can be used for revolution around of vertical axis changing the role of the variables.
    • Thread Starter
    Offline

    0
    ReputationRep:
    thanks for replying
    but i found there's another equation for the disc method in my textbook apart from the one above: int^b_a [f(x)]^2-[g(x)]^2 dx
    could you please tell me what is the difference?
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: March 19, 2011
Poll
“Yanny” or “Laurel”
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.