When a curve is given parametrically, its length is found by integrating the combined horizontal and vertical rates of change across the relevant parameter interval.

Using the arc length formula

A parametric curve is described by $x=f(t)$ and $y=g(t)$ on an interval $a\le t\le b$. To apply arc length correctly, the main task is identifying the correct interval and building the integrand from the two derivatives.

Arc length is the distance measured along the curve itself, not the straight-line distance between its endpoints.

For a parametrically defined curve, arc length is accumulated as the parameter moves from its starting value to its ending value.

This formula works directly from the parametric description. The needed information comes from the derivatives of the coordinate functions together with the interval of the parameter.

What the integrand means

The quantity inside the square root combines the horizontal and vertical changes of the curve.

It tells you how much length is being traced for a small change in $t$. If either coordinate changes quickly, the curve accumulates length more rapidly. If both change slowly, the length accumulates more slowly.

Because both derivatives are squared, the expression under the radical is never negative. That matches the idea that arc length is a distance and should not be negative.

Choosing the correct interval

The interval is part of the problem, not a minor detail.

In many questions, the curve is traced only over a specific range of parameter values, and the arc length must match that range exactly.

When applying the formula, check these points carefully:

• The interval should correspond to the part of the curve named in the problem.

• The curve may change direction as $t$ changes, but that does not change how the formula is used.

• If the same geometric path is traced more than once on the interval, the integral counts all of that travel.

This OpenStax figure depicts a semicircle traced by $x(t)=3\cos t$ and $y(t)=3\sin t$ over $0\le t\le \pi$, with the tracing direction marked. It reinforces that the arc length integral measures total distance traveled along the curve for the stated $t$-interval, so repeating or extending the parameter range would change the computed length. Source

• If the question asks for the length of a single tracing of a curve segment, the parameter bounds must reflect one tracing only.

A common error is choosing bounds by looking only at the rectangular coordinates. For parametric curves, the limits belong to the parameter, not directly to $x$ or $y$.

Reversing direction

A curve can be traced from end to start or from start to end. Its geometric length is the same either way, but the definite integral must be written with care. In practice, arc length is reported as a positive value, so it is safest to write the integral over the interval in increasing order unless the problem specifically requires another form.

Simplifying and evaluating the integral

After finding $dx/dt$ and $dy/dt$, square each derivative, add the results, and simplify before integrating. Careful algebra can turn a difficult-looking expression into a manageable one.

One subtle point is especially important: $\sqrt{(h(t))^2}=|h(t)|$, not automatically $h(t)$. If the integrand simplifies to the square root of a square, you must check whether an absolute value is needed. On a restricted interval, it may be possible to replace $|h(t)|$ with either $h(t)$ or $-h(t)$, but only after determining the sign of $h(t)$ on that interval.

Some arc length integrals can be evaluated exactly. Others do not have elementary antiderivatives and must be approximated numerically. On the AP Calculus BC exam, that may mean using calculator technology when instructed. Whether the answer is exact or approximate, include units when the problem provides them.

Conditions for using the formula

The formula is applied on an interval where the coordinate functions are differentiable and the integrand is defined. If the expression is not well behaved over the entire interval, it is often helpful to split the interval into smaller pieces and add the lengths.

In many standard AP problems, the curve is smooth enough that the formula can be applied directly. Even so, checking the derivatives before integrating is a good habit, because a correct formula with an incorrect interval or undefined expression will not give the intended length.

Common mistakes to avoid

• Using $\int_a^b \left(\dfrac{dx}{dt}+\dfrac{dy}{dt}\right),dt$ instead of the square-root formula

• Forgetting to square both derivatives before adding

• Mixing up parameter limits with coordinate values

• Dropping an absolute value during simplification

• Reporting a negative value for length

• Treating a repeated tracing as if it happened only once

• Leaving an answer as an indefinite integral when the problem asks for length over an interval

A reliable check is to ask whether the final answer makes sense as a distance: it should be positive, tied to the stated interval, and based on the full parametric behavior of the curve.