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.
Answer
We have
and we're given 0 ≤ t ≤ 4 so the length of the curve is
Example 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.
Answer
To graph this portion of the curve, we need to use t in the interval .
The derivatives are
so the length of the curve is
The calculator says
This portion of the curve should be longer than the straight-line distance but shorter than the block distance between
the points (0,2) and (1,0).
The straight-line distance is
and the block distance is
2 + 1 = 3.
Since our answer falls in between these, our answer is reasonable.
Example 3
Write an integral expression for the parametric function described by the equations
x(t) = x3
y(t) = etsin t
for 0 ≤ t ≤ π.
Answer
The derivatives of the parametric equations are
So the length of the curve is
We could simplify this, but it wouldn't get that much prettier so there's not much point.
Example 4
Write an integral expression for the length of the parametric curve
x = t cos t
y = t sin t
for α ≤ t ≤ β. Use the equality sin2t + cos2t = 1 to simplify your expression. (It will still be an integral.)
Answer
If
x = t cos t and y = t sin t
then
Squaring and adding the derivatives, some things cancel or simplify nicely:
We used the equality sin2t + cos2t = 1, and now we'll use it again:
.
The derivatives are 
