Integration/Applications of Integration2011. 6. 1. 15:33

Arc Length and Surfaces of Revolution


A rectifiable curve is one that has a finite arc length.
A sufficient condition for the graph of a function f to be rectifiable between (a,f(a)) and (b,f(b)) is
that f' be continuous on [a,b]. Such a function is continuously differentiable on [a,b], and
its graph on the interval [a,b] is a smooth curve.

\[ s = \int ds = \int_a^b \sqrt{1 + f^\prime(x)^2} ~dx \]


If the graph of a continuous function is revolved about a line, the resulting surface is
a surface of revolution.
The area A of the surface of revolution formed by rovolving the graph of f about a horizontal
or vertical axis is

\[ A = \int_a^b 2\pi f(x) \sqrt{1 + f^\prime(x)^2}~dx \]

where f(x) is the distance between the graph of f and the axis of revolution.

Usually, f(x) is equal to the distance between the graph of f and the axis of revolution,
like this example.



SurfaceOfRevolution(x^3,0..1,
'axis'='horizontal','distancefromaxis'
=0,'output'='plot');















But sometimes r(x) has a different value with f(x) like below problem.
It can be common mistake in solving surface area problems.



SurfaceOfRevolution(x^2,0..2^(1/2),
'axis'='vertical',
'distancefromaxis'=0,'output'='plot');
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Volume: The Washer Method


The disk method can be extende to cover solids of revolution with holes by replacing the representative disk with a representative washer.

To see how this concept can be used to find the volume of a solid of revolution, consider a region bounded by an outer radius Ro(x) and an inner radius Ri(x).

V= \pi \int_a^b \left({\left[R_O(x)\right]}^2 - {\left[R_I(x)\right]}^2\right) \mathrm{d}x

Here are examples which show how washers are adding up.



VolumeOfRevolution
(x^(1/2),x^2,0..1,'axis'=
'horizontal',
'distancefromaxis'=0,
'output'='plot');



VolumeOfRevolution
(x^(1/2),x^2,0..1,'axis'=
'horizontal',
'distancefromaxis'=0,
'output'='animation');


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Integration/Applications of Integration2011. 6. 1. 13:56

Volume: The Disk Method


If a region in the plane is revolved about a line, the resulting solid is a solid of revolution,
and the line is called the axis of revolution.

To see how to use the volume of a disk to find the volume of a general solid of revolution, consider a solid of revolution formed by revolving the plane region about the indicated axis.

V= \pi \int_a^b {\left[R(x)\right]}^2\ \mathrm{d}x

Here are examples that explain how this method works.



VolumeOfRevolution(sqrt(sin(x)),
0,0..π,'axis'='horizontal',
'distancefromaxis'=0,output=
plot)
















VolumeOfRevolution(sqrt(sin(x)),
0,0..π,'axis'='horizontal',
'distancefromaxis'=0,output=
animation)













This figure shows adding disks in the general solid, which is the basic concept of the disk method.
As divided terms are decreasing to zero, or subinterval n is increasing to infinity,
it makes the formula above.


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Approximation of Integration (The Trapezoidal Rule)


The Trapezoidal Rule

It is the method of approximation a difinite integral with using trapezoids.
Assume that f is ontinuous and positive on the interval [a,b].
So, the definite integral f(x) a to b as x values represents the area of the region bounde by
the graph of f and the x-axis, from x=a to x=b.
By adding these trapezoids up, you can approximate integrations.

(1/2)*(b-a)*(f(x[0])+2*f(x[1])+2*f(x[2])+`...`+2*f(x[N-1])+f(x[N]))/N

Here are examples.



with(Student[Calculus1]);
>ApproximateInt(sin(x),x=0..π, method=trapezoid, output=plot)















As n, the number of subintervals, is going to infinity,
approximation of an integral approaches its real value.  



with(Student[Calculus1]);
>ApproximateInt(sin(x),x=0..π, method=trapezoid, output=animation)

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Approximation of Integration (The Lower Sums and the Upper Sums)

The sum of the areas of the inscribed rectangles is called a lower sum.
The sum of the areas of the circumscribed rectangles is called an upper sum.

Here is the example.

Find the upper and lower sums for the region bounded by the graph of f(x) = x^2 and the x-axis between x=0 and x=2.

The Lower Sum, using the left endpoints (Figure 2-01)


with(student);
>f := x -> x ;
>leftbox(f(x), x = 0 .. 2);















The Upper Sum, using the right endpoints (Figure 2-02)


with(student);
>f := x -> x ;
>rightbox(f(x), x = 0 .. 2);












 
 


Limits of the Lower and Upper Sums

Let f be continuous and nonnegative on the inverval [a,b]. When n is going to infinity, the limits of both the lower and upper sums exist and are equal to each other. 


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