ds vs. dx: When Do We Use Each One?

A Q&A Guide to Area and Arc Length in Parametric Curves

When studying parametric curves, it is easy to confuse

.

All three are written using the familiar differential notation, but they represent different ideas.

A particularly important question is:

If represents a small piece of the curve, why don’t we use when calculating the area under the curve?

Let’s answer this step by step.

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Q1. What does (ds) represent?

Suppose a point moves along a curved path.

The tiny distance travelled along the curve is called .

If the movement has a horizontal component and a vertical component , then by the Pythagorean theorem,

.

Therefore,

.

So represents:

a tiny amount of actual distance along the curve.

This makes naturally useful when we want to calculate arc length.

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Q2. How is (ds) used to find arc length?

If the curve consists of many tiny pieces , then adding all those pieces gives the total length:

.

For a parametric curve

,

we have

and

.

Consequently,

.

Therefore,

.

So for arc length:

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Q3. Why don’t we use (ds) to calculate the area under the curve?

Because the area calculation is based on a different type of small element.

For a curve , imagine dividing the area into very thin vertical rectangles.

Each rectangle has:

height:

width:

Therefore, its small area is

.

Adding all those small areas gives

.

Notice that the width of the rectangle is , not .

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Q4. What is the geometric difference between (dx) and (ds)?

Consider a tiny portion of a curve:

•
                /|
               / |
              /  | dy
             /   |
            •----•
                dx

        slanted side = ds