# FP2 June 05 Surface AreaWatch

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#1
A curve is described by

The curve is rotated through 2pi radians about the x-axis. Find the exact value of the area of the curved surface generated.

I've done the integration and all that but I'm confused about the limits I should use. A picture of the curve is attached. When it cuts the x axis, y = 0, so

asin^3t = 0
sin^3t = 0
sint = 0
t = 0 (pi is not a solution)

And the curve cuts the y axis when x = 0, so

cost = 0
t = pi/2

Then shouldn't the limits of the integral be ? Why is it the other way round? I'm sure it's something obvious that I'm missing, it is 6 AM after all....
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11 years ago
#2
Hmm. I've done this Q successfully. Hangon.
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11 years ago
#3
My GUESS is...

The formula for the cartesian curved surface area is ... well, the one with in it (sorry my Latex skills are bad!)

The limits for that will be in increasing order e.g. 0 and 1.

When you derive the formula for the same thing in parametric form, it's like a substitution for t instead of x, and with a substitution comes a change in limits, so x value goes to t value and this may warrant a swap-over to decreasing order.

Well... I might be being stupid, it's 6.30AM or so after all!
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11 years ago
#4
No, thats not how you work out the limits.

The limits are given in the question. Upper bound pi/2, lower bound 0.
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11 years ago
#5
Yes, but it hasn't stated which is the upper and which is the lower bound.
Anyway, this is a stab in the dark, it does seem odd but if you're going in order of increasing x values then the t will reverse its values in this case.
I'm guessing that the formula for curved surface area is derived in cartesian form first (though I may be wrong) and so when specifying the limits it goes in order of increasing x values, not t values.

Edit: Thinking about it more, and I'm pretty sure that's it.
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11 years ago
#6
(Original post by ahezhara)
Yes, but it hasn't stated which is the upper and which is the lower bound.
Anyway, this is a stab in the dark, it does seem odd but if you're going in order of increasing x values then the t will reverse its values in this case.
I'm guessing that the formula for curved surface area is derived in cartesian form first (though I may be wrong) and so when specifying the limits it goes in order of increasing x values, not t values.

Edit: Thinking about it more, and I'm pretty sure that's it.
It doesnt have to state it;

Definition: If a function y = f(x) has a continuous first derivative throughout the interval a < x < b, then the area of the surface generated by revolving the curve about the x-axis is; S = 2pi INT y(1+(dy/dx)^2)^(1/2).dx [from a to b]

source: http://curvebank.calstatela.edu/arearev/arearev.htm
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11 years ago
#7
, and as area is scalar, it shouldn't make a big difference.
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11 years ago
#8
(Original post by Swayum)
A curve is described by

The curve is rotated through 2pi radians about the x-axis. Find the exact value of the area of the curved surface generated.

I've done the integration and all that but I'm confused about the limits I should use. A picture of the curve is attached. When it cuts the x axis, y = 0, so

asin^3t = 0
sin^3t = 0
sint = 0
t = 0 (pi is not a solution)

And the curve cuts the y axis when x = 0, so

cost = 0
t = pi/2

Then shouldn't the limits of the integral be ? Why is it the other way round? I'm sure it's something obvious that I'm missing, it is 6 AM after all....
It's given in parametric form, so use

Everything must be in terms of (including , and the limits). The limits are in terms of , not , and so you must use the limits , .

The formula I gave is in the formula book. If it's parametric, make sure you go with parametric.

The way I do these questions is I work out either what is (if it's cartesian, not parametric) or I work out what is, if it is given in parameters.
In the question given, I found this to be (after fumbling around, cancelling down, manipulation, etc), and so I then multiplied this by and thus the integral becomes which is easy to solve.
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11 years ago
#9
Okay for this question the limits are given in the question, and 0.
Its a standard result, you just need the formula, which is the integral of
It should work from there.

Btw, those values are the "t" values. If you used a sub like I did, then in the end I got limits of 0 and 1. So it should be clear where each one goes.
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