**Introduction Arc Length and Surface Area**

Just as the integral may be used to calculate planar area and spatial volume, so this tool may also be used to calculate the arc length of a curve and surface area. The basic idea is to approximate the length of a curve by the length of its piecewise linear approximation. A similar comment applies to the surface area. We begin by describing the basic rubric.

**Arc Length**

Suppose that *f* (*x*) is a function on the interval [ *a, b* ]. Let us see how to calculate the length of the curve consisting of the graph of *f* over this interval (Fig. 8.28). We partition the interval:

*a* = *x* _{0} ≤ *x* _{1} ≤ *x* _{2} ≤ ... ≤ *x* _{k −1} ≤ *x _{k}* =

*b*.

*l _{j}* = ([

*x*−

_{j}*x*

_{j −1}]

^{2}+ [

*f*(

*x*) −

_{j}*f*(

*x*

_{j −1})]

^{2})

^{1/2}.

We denote the quantity *x _{j} − x*

_{j −1 }by Δ

*x*and apply the definition of the derivative to obtain

Now we may rewrite the formula for *l _{j}* as

Summing up the lengths *l _{j}* (Fig. 8.30) gives an approximate length for the curve:

But this last is a Riemann sum for the integral

As the mesh of the partition becomes finer, the approximating sum is ever closer to what we think of as the length of the curve, and it also converges to this integral. Thus the integral represents the length of the curve.

**Examples**

**Example 1 **

Let us calculate the arc length of the graph of *f* ( *x* ) = 4 *x* ^{3/2} over the interval [0,3].

**Solution 1 **

The length is

**Example 2 **

Let us calculate the length of the graph of the function *f* ( *x* ) = (1/2) × (*e ^{x}* +

*e*) over the interval [1, ln 8].

^{−x}**Solution 2 **

We calculate that

*f′* (*x*) = (1/2)(*e ^{x} − e ^{−x}*).

Therefore the length of the curve is

**You Try It**: Set up, but do not evaluate, the integral for the arc length of the graph of on the interval π/4 ≤ *x* ≤ 3π/4.

Sometimes an arc length problem is more conveniently solved if we think of the curve as being the graph of *x* = *g* (*y*). Here is an example.

**Example**

Calculate the length of that portion of the graph of the curve 16 *x* ^{2} = 9 *y* ^{3} between the points (0,0) and (6,4).

**Solution **

We express the curve as

Then . Now, reversing the roles of *x* and *y* in (*), we find that the requested length is

This integral is easily evaluated and we see that it has value [2 · (97) ^{3/2} − 128]/243.

Notice that the last example would have been considerably more difficult (the integral would have been harder to evaluate) had we expressed the curve in the form *y* = *f* (*x*).

**You Try It**: Write the integral that represents the length of a semi-circle and evaluate it.

Find practice problems and solutions for these concepts at: Applications of the Integral Practice Test.

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