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integration for finding volume

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
Reply 1
Avatar for eliotball
eliotball
7
In general when a curve is rotated around the x-axis between x = a and x = b the volume is
∫abπy2dx=π∫aby2dx\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:
Reply 2
Avatar for ztibor
ztibor
10
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 Δx\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 Δx→dx\Delta x \rightarrow dx and summing the elementary volumes we get
V=π⋅∫ab(R(x))2dxV=\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 Δy\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 Δy→dy\Delta y \rightarrow dy and summing we get the volumes as integral of
V=2π∫y1y2(y−y0)⋅(g(y2)−g(y)) dyV=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.
(edited 13 years ago)
Reply 3
Avatar for kingsclub
kingsclub
OP
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?

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