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Area, Volume, and Arc Length

Area, Volume, and Arc Length

At a Glance - Arc Length for Parametric Functions

We just learned how to chop up a simple herb like a chive. They're just simple lines that we know how to describe in terms of the variables x and y. Sometimes, we need to cut up more intricate herbs like rosemary. These lines may be described in different ways, including in parametric equations.

These parametric equations are just a different language for describing lines. They're nice because it's easier to describe the curve using this other language, and it can be easier to integrate this way too.

The process for finding the length of a curve described by parametric equations is the same. We're just using a different language. It's not even an African click language.

Suppose we want to find the length of the curve described by parametric equations x(t) and y(t), on the interval atb.

We break up the curve into little pieces, as usual:

The Pythagorean Theorem says that the length of a little piece is (approximately)

We need to get Δ x and Δ y in more useful terms, preferably in terms of t, so we can end up with an integral with respect to t.

The amount x changes over a little time interval is approximately  multiplied by the length of the time interval.

Similarly, for y,

This means the length of a little piece is approximately

This simplifies to

When we integrate over the values that make sense for t, we get the length of the curve:

This equation is just like the one we derived in the previous section, but now the equation is parameterized with variable t. We're using the notation  instead of the notation y'(t) to make it clear that t is the independent variable. Since there are so many variables floating around, using prime notation here isn't the best idea. It's best not to get confused with dots and quotation or prime marks floating around a page of mathematical soup.

Here is a recap video of Pythagorean Theorem, just in case you need it.

Sample Problem

Find the length of the parametric curve described by

x(t) = t2, y(t) = t3

for 0 ≤ t ≤ 2.

You may use a calculator to evaluate the integral.

Explain why the answer is reasonable.


First, we find the derivatives

Using our new formula, the length of the curve is

If we graph the curve, we get

This curve is close enough to the straight line between (0,0) and (4,8), so we would expect the length of the curve to be close to the length of that line.

The length of the line is

Since that's close to our answer, our answer makes sense.

When we're asked to explain why an answer is reasonable, we should compare the curve to a line or circle whose length we know how to find. Even when we're not explicitly asked to do this, it's a good idea as a way to check ourselves. If a curve is as long as a green bean, we don't want to trust an answer that says it's as long as Route 66.

With parametric functions we have to take a little extra care with the limits of integration. The limits of integration are values of t. If we aren't given the values of t, we have to find them ourselves.

Fun fact: "normal" functions are just a special case of parametric functions. The function f(x) can be parametrized as

x(t) = t

y(t) = f(t).

Since x and t are the same thing, the curve f(x) for axb is the same as the parametrized curve for atb. We can find the length of the curve by working with the parametrized version.

The derivative of

x(t) = t is

and the derivative of y(t) = f(t) is

Then the length of the curve is

Since x and t are the same, we could just as well write this as

which is the same thing the formula we had earlier.

This means we don't need to remember multiple formulas for arc length. Having to memorize less is better. It'll keep us from making mistakes between formulas, or from mistaking a panda for a polar bear.

If we remember the version for parametric equations, we can find the arc length of a normal function by using the parametrization x = t.

We prefer the parametric version. It looks just a bit more complicated, but it's more fun. The integral

Looks like it's using the Pythagorean Theorem, which makes it easier to remember.

Example 1

The parametric equations

x(t) = cos 2t, y(t) = sin 2t

trace the unit circle. Use the arc length formula to find the circumference of the unit circle.

Exercise 1

Write an integral expression for the length of the curve described by the parametric equations x(t) = et and y(t) = 2t + 1 for 0 ≤ t ≤ 4.

Exercise 2

The equations x(t) = cos t and y(t) = 2sin t describe a parametric curve.

(a) Write an integral expression for the length of the portion of the curve graphed below.

(b) Use a calculator to evaluate your integral and explain why your answer is reasonable.

Exercise 3

Write an integral expression for the parametric function described by the equations

x(t) = x3

y(t) = etsin t

for 0 ≤ t ≤ π.

Exercise 4

Write an integral expression for the length of the parametric curve

x = t cos t

y = t sin t

for αtβ. Use the equality sin2 t + cos2 t = 1 to simplify your expression. (It will still be an integral.)

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