Difference between revisions of "Arc Length and Surface Area"

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'''Example'''
 
'''Example'''
: Length of the curve <math>y=2x</math> from <math>x=0</math> to <math>x=1</math>}}
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: Length of the curve <math>y=2x</math> from <math>x=0</math> to <math>x=1</math>
 
As a sanity check of our formula, let's calculate the length of the "curve" <math>y=2x</math> from <math>x=0</math> to <math>x=1</math> . First let's find the answer using the Pythagorean Theorem.
 
As a sanity check of our formula, let's calculate the length of the "curve" <math>y=2x</math> from <math>x=0</math> to <math>x=1</math> . First let's find the answer using the Pythagorean Theorem.
 
:<math>P_0=(0,0)</math>
 
:<math>P_0=(0,0)</math>

Latest revision as of 16:58, 15 January 2022

Arc Length

Suppose that we are given a function that is continuous on an interval and we want to calculate the length of the curve drawn out by the graph of from to . If the graph were a straight line this would be easy — the formula for the length of the line is given by Pythagoras' theorem. And if the graph were a piecewise linear function we can calculate the length by adding up the length of each piece.

The problem is that most graphs are not linear. Nevertheless we can estimate the length of the curve by approximating it with straight lines. Suppose the curve is given by the formula for . We divide the interval into subintervals with equal width and endpoints . Now let so is the point on the curve above . The length of the straight line between and is

So an estimate of the length of the curve is the sum

As we divide the interval into more pieces this gives a better estimate for the length of . In fact we make that a definition.

Length of a Curve

The length of the curve for is defined to be

The Arclength Formula

Suppose that is continuous on . Then the length of the curve given by between and is given by

And in Leibniz notation

Proof: Consider . By the Mean Value Theorem there is a point in such that

So

Putting this into the definition of the length of gives

Now this is the definition of the integral of the function between and (notice that is continuous because we are assuming that is continuous). Hence

as claimed.

Example

Length of the curve from to

As a sanity check of our formula, let's calculate the length of the "curve" from to . First let's find the answer using the Pythagorean Theorem.

and

so the length of the curve, , is

Now let's use the formula

Exercises

1. Find the length of the curve from to .

2. Find the length of the curve from to .

Arclength of a parametric curve

For a parametric curve, that is, a curve defined by and , the formula is slightly different:

Proof: The proof is analogous to the previous one: Consider and .

By the Mean Value Theorem there are points and in such that

and

So

Putting this into the definition of the length of the curve gives

This is equivalent to:

Exercises

3. Find the circumference of the circle given by the parametric equations , , with running from to .

4. Find the length of one arch of the cycloid given by the parametric equations , , with running from to .

Exercise Solutions

Surface Area

Suppose we are given a function and we want to calculate the surface area of the function rotated around a given line. The calculation of surface area of revolution is related to the arc length calculation.

If the function is a straight line, other methods such as surface area formulae for cylinders and conical frusta can be used. However, if is not linear, an integration technique must be used.

Recall the formula for the lateral surface area of a conical frustum:

where is the average radius and is the slant height of the frustum.

For and , we divide into subintervals with equal width and endpoints . We map each point to a conical frustum of width Δx and lateral surface area .

We can estimate the surface area of revolution with the sum

As we divide into smaller and smaller pieces, the estimate gives a better value for the surface area.

Definition (Surface of Revolution)

The surface area of revolution of the curve about a line for is defined to be

The Surface Area Formula

Suppose is a continuous function on the interval and represents the distance from to the axis of rotation. Then the lateral surface area of revolution about a line is given by

And in Leibniz notation

Proof:

As and , we know two things:

  1. the average radius of each conical frustum approaches a single value
  2. the slant height of each conical frustum equals an infitesmal segment of arc length

From the arc length formula discussed in the previous section, we know that

Therefore

Because of the definition of an integral , we can simplify the sigma operation to an integral.

Or if is in terms of on the interval


Resources

Arc Length


Surface Area

Licensing

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