The Area of a Surface of Revolution
by Steve Hennagin with lots of help from Mike May
Copyleft 2007
A surface of revolution is found by rotating a curve F(x, y) = 0 about a line \356\223\201. The general problem is difficult
to implement so we will concentrate on curves of the form y=f(x) or x = g(y) with rotation about a line parallel
to one of the coordinate axes. The simplest curve to rotate is the constant curve.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic=
f := x -> 4;
JSFH a := 0;
b := 7;
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlIiIlRiVGJUYl
IiIh
IiIo
JSFH
In this case we can simpply use the formula for the surface area of the side of a cylinder:
QyQ+SSxhcmVhc3VyZmFjZUc2ImYqNiRJInJHRiVJImhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwkKihJI1BpRyUqcHJvdGVjdGVkRyIiIjkkRjE5JUYxIiIjRiVGJUYlRjE=
Zio2JEkickc2IkkiaEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKEkjUGlHJSpwcm90ZWN0ZWRHIiIiOSRGLjklRi4iIiNGJUYlRiU=
In our example,
QyY+SSdyYWRpdXNHNiIiIiUiIiI+SSdoZWlnaHRHRiUiIihGJw==
IiIl
IiIo
Which gives
QyQvLi1JLGFyZWFzdXJmYWNlRzYiNiRJJ3JhZGl1c0dGJ0knaGVpZ2h0R0YnRiUiIiI=
Ly1JLGFyZWFzdXJmYWNlRzYiNiRJJ3JhZGl1c0dGJUknaGVpZ2h0R0YlLCRJI1BpRyUqcHJvdGVjdGVkRyIjYw==
JSFH
Now, we will compute the value as an integral:
First, we plot the curve.
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
JSFH
Next we parametrize the curve so that it can be parameterized.
Qyg+SSVzdXJmRzYiNyVJInhHRiUqJi1JImZHRiU2I0YnIiIiLUkkc2luR0YlNiNJInRHRiVGLComRilGLC1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYvRixGLD5JI3QwR0YlIiIhRiw+SSN0MUdGJSwkSSNQaUdGNSIiI0Ys
NyVJInhHNiIsJC1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJ0R0YkIiIlLCQtSSRjb3NHRihGK0Yt
IiIh
LCRJI1BpRyUqcHJvdGVjdGVkRyIiIw==
Now rotate:
Error, missing operator or `;`
plot3d(surf, x=a..b, t=t0..t1,
axes=boxed, scaling=constrained);
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The integral for computing the area of a surface of revolution is
The area of a surface of revolution is the integral of
2Pi*(f(x))LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 over the domain
JSFH
QyQ+SSNmeEc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJmR0YlNiM5JEYzRiVGJUYlIiIi
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklZGlmZkclKnByb3RlY3RlZEc2JC1JImZHRiU2IzkkRjBGJUYlRiU=
Since the function is constant, its derivative is zero. The integral is thus
QyQvLUkkSW50RzYiNiQsJCooSSNQaUclKnByb3RlY3RlZEciIiItSSVzcXJ0RzYkRitJKF9zeXNsaWJHRiY2IywmRixGLCokLUkjZnhHRiY2I0kieEdGJiIiI0YsRiwtSSJmR0YmRjZGLEY4L0Y3O0kiYUdGJkkiYkdGJi1JJGludEdGJkYnRiw=
Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCRJI1BpR0YmIiIpL0kieEdGKDsiIiEiIigsJEYrIiNj
JSFH
We now follow this with the example of a semicircle of radius one rotated about the x-axis, the answer ought to be 2* Pi, which is one-half the surface area of the unit sphere. Let us see if it is so.
Qyg+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYiIiJGMyokOSQiIiMhIiJGJUYlRiVGMz5JImFHRiUiIiFGMz5JImJHRiVGM0Yz
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYiIiJGMCokOSQiIiMhIiJGJUYlRiU=
IiIh
IiIi
QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSJmR0YoNiNJInhHRigvRi07SSJhR0YoSSJiR0YoIiIi
NigtJSdDVVJWRVNHNiQ3WTckJCIiISEiIiQiIzUhIiI3JCQiMm1tbW1UOih6QCEjPSQiMSk9OiJRVGkoKioqISM7NyQkIjJOTExlOXVpMiUhIz0kIjF3Z19TJilvIioqKiEjOzckJCIxbm1tInpfIjRpISM8JCIxJ1xzXmYvMikqKiEjOzckJCIxbW1tVCZwaE4pISM8JCIxQCJmazBFXScqKiEjOzckJCIyS0wkZSo9KUhcNSEjPCQiMVxOKyRIJ3pXKiohIzs3JCQiMm1tO3ovM3VDIiEjPCQiMW00JHlpJCo9IyoqISM7NyQkIjIpKioqXDdMUkRYIiEjPCQiMUZ6bSIzV1IqKSohIzs3JCQiMmxtO3pSJ29rOyEjPCQiMWJhIlFcbi8nKSohIzs3JCQiMikqKipcaTVgaCg9ISM8JCIxTW01WmVVQSkqISM7NyQkIjJOTEwkM0VuJDQjISM8JCIxL0F5dTJQeSgqISM7NyQkIjJubW1UIVJFJkcjISM8JCIxSkJrIz14YHQqISM7NyQkIitELiY0XSMhIzUkIjBaNkglSEAjbyohIzo3JCQiK3ZCXzxGISM1JCIxX15BKFtzT2kqISM7NyQkIit2J0hpI0ghIzUkIjE7I2U3LHpBYyohIzs3JCQiMm1tO3oqZXY6SiEjPCQiMUFcOypwOEFdKiEjOzckJCIxTExMMzQ3VEwhIzskIjFYIUdIWkxgVSohIzs3JCQiMUxMTExZLktOISM7JCIxaVN1PmBZYiQqISM7NyQkIi1EIm83VHYkISM3JCIxdWFqJlEkZW8jKiEjOzckJCIxTExMJFEqb11SISM7JCIxYD8neXY5bD0qISM7NyQkIi1EIj1sajslISM3JCIxbHhVMzt0ITQqISM7NyQkIi12ViZSPFAlISM3JCIxMzQlcHBwUCoqKSEjOzckJCIyTkxMZTlFZ2UlISM8JCIxRVh1JmU5aykpKSEjOzckJCIyTEwkZVIiM0d5JSEjPCQiMScpKUg0ZnA/eSkhIzs3JCQiMm1tbVQ1a10qXCEjPCQiMW0kelAhPjVqJykhIzs3JCQiMW1tInpSUWJAJiEjOyQiMTEmPTV0JD1LJikhIzs3JCQiLXY9PlkyYSEjNyQiMWUmeVouZT1UKSEjOzckJCIxbW07elh1OWMhIzskIjEvWEAkb1NcRikhIzs3JCQiKiZ5KSlHZSEiKiQiMSJ5U2hYPmI3KSEjOzckJCIxKioqKlxpX1FRZyEjOyQiMW5NbFVtMXJ6ISM7NyQkIjEqKipcN3klM1RpISM7JCIxWSR6KlFxUDh5ISM7NyQkIjEqKioqXFAhW2hZJyEjOyQiMVhoSWFvO0d3ISM7NyQkIjFMTEwkUXgkb20hIzskIjEjcFwrWUg/WCghIzs3JCQiK3YuSSUpbyEjNSQiMThJckh2LWBzISM7NyQkIjFtbSJ6cGUqenEhIzskIjF5bm5LZDtpcSEjOzckJCIrRFwnUUgoISM1JCIxIlFUWl89NSVvISM7NyQkIjFLTGU5UzgmXCghIzskIjEmUnAoKXAiKik+bSEjOzckJCIxKioqXGk/PWJxKCEjOyQiMTxIbVVOeHRqISM7NyQkIjFMTEwzcz82eiEjOyQiMXBoUC8vZztoISM7NyQkIi1ESlhhRSIpISM3JCIxOVI9QCdldSNlISM7NyQkIjFubW1tKlJSTCkhIzskIjBuMjNNJHpFYiEjOjckJCIxbW07YTwuWSYpISM7JCIwbU5FSjtHPiYhIzo3JCQiMUxMZTl0T2MoKSEjOyQiMnVPVShSSXFIWyEjPDckJCIqJlFrXCopISIqJCIxQEk6L2ZQaFchIzs3JCQiMUxMJDNkZzY8KiEjOyQiMlgoKSo0R0g/JylSISM8NyQkIjFtbW1teEdwJCohIzskIjEiUkw2Ui5fXCQhIzs3JCQiLUQib0swZSohIzckIjE9Q2lzPSFmJ0chIzs3JCQiKiZvaSJvKiEiKiQiMVVKMU8jPUtdIyEjOzckJCItdj01cyN5KiEjNyQiMXZPPlU0RHQ/ISM7NyQkIi9EMWsyL1ApKiEjOSQiMlBWdWVMXnp6IiEjPDckJCIudiQ0ME8iKikqISM4JCIxSC5mJW9IK1oiISM7NyQkIjFeNy4jUT8mPSoqISM7JCIyQj56Xio9JlJGIiEjPDckJCIvdm9hLW9YKiohIzkkIjFQJVw4SCcpMy8iISM7NyQkIjB2VnQ3U0coKiohIzokIjEmPWhKKnA9bHQhIzw3JCQiIzUhIiIkIiIhISIiLSUmQ09MT1JHNiYlJFJHQkckIiM1ISIiJCIiISEiIiQiIiEhIiItJSVWSUVXRzYkOyQiIiEhIiIkIiM1ISIiJShERUZBVUxURy0lK0FYRVNMQUJFTFNHNidJInhHNiJRITYiLSUlRk9OVEc2JSUhRyUhRyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUqQVhFU1NUWUxFRzYjJSdOT1JNQUxHLSUoU0NBTElOR0c2IyUuVU5DT05TVFJBSU5FREctJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRJJCEiIi0lKUJPVU5EU19ZRzYjJCIkPyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVJTiEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiVnTSEiIi0lKUNISUxEUkVORzYi
Qyg+SSVzdXJmRzYiNyVJInhHRiUqJi1JImZHRiU2I0YnIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJInRHRiVGLComRilGLC1JJGNvc0dGL0YyRixGLD5JI3QwR0YlIiIhRiw+SSN0MUdGJSwkSSNQaUdGMCIiI0Ys
NyVJInhHNiIqJiwmIiIiRicqJEYjIiIjISIiI0YnRiktSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kidEdGJEYnKiZGJkYrLUkkY29zR0YuRjFGJw==
IiIh
LCRJI1BpRyUqcHJvdGVjdGVkRyIiIw==
Draw the surface of revolution:
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JSFH
Finally we put it all together in one place. The student only need change the first few values to fit the problem at hand.
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IiIh
IiIi
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Do the plotting:
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IiIh
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JSFH
Integrate:
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JCIrMChIKDRRISIq
JSFH
JSFH
Here it is, all together in one package.
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IiIi
IiIk
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Exercises:
Find the area of the follwoing surfaces of revolution
1. f(x) = sin(x), x = Pi/3 .. 2 * Pi/3, about the x-axis
2. f(x) = e^(x^2), x = 1.. 3, about the line y = -1 (Extra Credit)
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