Calc1231W13T4 - Surface area of revolution

Sunday, 21 October 2018

7:19 PM

Untitled picture.png The surface figured, which is part of a right-circular cone, is called a frustrum
frustrum
But it kind of looks like a fez
fez
If the radii of the top and bottom are r and R and the slant height is s (note lower case s), then the surface area of the frustrum (not including the base or top) is
So to make a fez with top radius 9 cm, bottom radius 10 cm and slant height 17 cm requires
O squared cm of felt.
Note: the Maple syntax for T is pi and to make a fez requires enough felt to cover the frustrum and the top circle.
Untitled picture.png Machine generated alternative text:
The function f(x)
e
for x
O to oo can be rotated about the x axis to form a surface of revolution S, which looks like a traffic cone (if you chop off the infinite tail')
Surface of revolution S
Traffic Cones
The Duke of Wellington, by Patrick Morgan CC-BY 2.0
In general; the surface area of a surface of revolution about the x -axis of a curve described by y = f(x) with a < x < b, is given by the formula
b
A = 27Tf(x) 1 + (f' dc
Find the surface area of the surface S:
This integral may help with your working,
1
1+u2du-
Note: the Maple notation for is pi. The Maple syntax for log(x) is In (x) If you would like to know more, click here to see a video that nicely demonstrates the process by which surfaces
of revolution are formed.
Untitled picture.png 10) •17 + pi.92
04

Untitled picture.png Machine generated alternative text:
The function f(x)
e
for x
O to oo can be rotated about the x axis to form a surface of revolution S, which looks like a traffic cone (if you chop off the infinite tail')
Surface of revolution S
Traffic Cones
The Duke of Wellington, by Patrick Morgan CC-BY 2.0
In general; the surface area of a surface of revolution about the x -axis of a curve described by y = f(x) with a < x < b, is given by the formula
b
A = 27Tf(x) 1 + (f' dc
Find the surface area of the surface S:
This integral may help with your working,
1
1+u2du-
Note: the Maple notation for is pi. The Maple syntax for log(x) is In (x) If you would like to know more, click here to see a video that nicely demonstrates the process by which surfaces
of revolution are formed.

Untitled picture.png A curve can also be specified using polar co-ordinates and then rotated about the x-axis. Consider the curve specified by r
1 where O < 9 < -L _ It looks a bit like a knit cap.
Surface of revolution S
Knit Cap
In general, the surface area of a surface of revolution about the x -axis of a curve described in polar co-ordinates by r = f(9) with 90 S 9 91 is
2m sin(9) r +
Hence the surface area of S is
Note: the Maple notation for is pi.
Untitled picture.png Machine generated alternative text:
: proc(fg b)
int(2 Pi sqrt(l
end proc
A (exp( -x), O, infinity)
proc(fg b) +
end proc

Untitled picture.png Machine generated alternative text:
: proc(fg b)
int(2 Pi sqrt(l
end proc
A (exp( -x), O, infinity)
proc(fg b) +
end proc

$Untitled picture.png A curve can also be specified using polar co-ordinates and then rotated about the x-axis. Consider the curve specified by r 1 where O < 9 < -L _ It looks a bit like a knit cap. Surface of revolution S Knit Cap In general, the surface area of a surface of revolution about the x -axis of a curve described in polar co-ordinates by r = f(9) with 90 S 9 91 is 2m sin(9) r + Hence the surface area of S is Note: the Maple notation for is pi. Untitled picture.png A curve can also be specified in parametric form and then rotated about the x-axis. Consider C : {[t, cosh(t)], 0 < t < 1} rotated about the x-axis. It looks a bit like a stovepipe. Surface of revolution S Stovepipe In general, the surface area of a surface of revolution about the x -axis for a curve described in parametrically by [x(t), y(t)] for a < t < b is A Hence the surface area of S is Note: the Maple notation for is pi. dc 27TY(t) dt dt Untitled picture.png A : —int (t=t , y) *sqrt (diff (x , t) subs (a=O, b=l, A) ; simplifi %) A2 + diff (y, t) A2) , = 7tsinh(b) 1 7csinh(1) 1 sinh(a) 1 sinh(O) 1 (cosh(l) sinh(l) 1)$

Untitled picture.png A : —int (t=t , y) *sqrt (diff (x , t)
subs (a=O, b=l, A) ;
simplifi %)
A2 + diff (y, t)
A2) ,
= 7tsinh(b) 1
7csinh(1) 1
sinh(a) 1
sinh(O) 1
(cosh(l) sinh(l)
1)