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## Main Question or Discussion Point

Several authors state the formula for finding the arc length of a curve defined by ##y = f(x)## from ##x=a## to ##x=b## as:

$$\int ds = \int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx$$

Isn't this notation technically wrong, since the RHS is a definite integral, and the LHS is an indefinite integral (family of antiderivatives)?

I understand that no limits are placed on the integral of ##ds## since ##ds## can be defined in several equivalent ways: y as the independent variable, x as the independent variable, parametrically, or in terms of polar coordinates.

But why can't we write the integral as:

$$\int_{x=a}^{x=b} ds$$

Writing it this way makes it explicit that the variable ##x## is changing from ##a## to ##b##.

Alternatively, can't we also write it as:

$$\int_P ds$$

Where P is the path I have defined above.

$$\int ds = \int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx$$

Isn't this notation technically wrong, since the RHS is a definite integral, and the LHS is an indefinite integral (family of antiderivatives)?

I understand that no limits are placed on the integral of ##ds## since ##ds## can be defined in several equivalent ways: y as the independent variable, x as the independent variable, parametrically, or in terms of polar coordinates.

But why can't we write the integral as:

$$\int_{x=a}^{x=b} ds$$

Writing it this way makes it explicit that the variable ##x## is changing from ##a## to ##b##.

Alternatively, can't we also write it as:

$$\int_P ds$$

Where P is the path I have defined above.

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