Educator

Parametric Curves

Main formula:

Slope m =

Arclength =

Hints and tips:

• You can remember the formula for the slope of the tangent line by thinking that symbolically, the dt’s cancel, leaving you with dy ⁄ dx.

• To find the equation of the tangent line, you also need a point. Use the given value of t into x(t) and y(t) to find it. Then you can use the point-slope formula from high school algebra (y − y₀ = m(x − x₀ )) to find the equation.

• Sometimes you aren’t given a value of t, but the coordinates (x, y) instead. Then you must find which value of t gives you the correct (x(t), y(t)). Make sure you check that both x and y are correct for your value of t.

• You can remember the arclength formula by recalling that it is derived from the distance formula between two points, which in turn comes from the Pythagorean Theorem.

• Don’t make the common algebraic mistake of thinking that reduces to a + b! This is extremely wrong, and your teacher will likely be merciless if you do it

• Many problems in Calculus II classes are “rigged” so that when you expand x′(t)² + y′(t)² , it becomes a perfect square that cancels nicely with the square root.

• Often this perfect square is achieved by making one of x′(t)² and y′(t)² be something of the form (a − b)² = a² − 2ab + b². Then the other one changes it to a² + 2ab + b², which you can then factor as (a + b)².

• Another common technique in arclength problems is to make a u-substitution for whatever is under the square root sign. Then (hopefully) you can manipulate the expression outside the square root into being the du. However, you might have to do several steps of algebraic manipulation, pulling factors in or out of the square root sign, before this works.

• You may sometimes be able to use symmetry to find the arclength of part of a curve and then multiply by an appropriate factor to get the total arclength. This can be especially helpful if you just want to examine part of the curve where all the quantities involved are positive.

• When it’s feasible, check that your answer makes sense. Unlike area integrals, which can be negative if a curve goes below the x-axis, arclength should always be positive! You might also be able to check that the curve looks about as long as your answer.