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 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)

 

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