BC 2-3
Arc Length
Name:
(1) Suppose that we have a curve in the plane given by the parametric equations:
x  x(t ), y  y (t ), a  t  b . Find a formula for the length of the curve. You may
assume that the curve is smooth. Remember that a parametric curve is smooth if and
only if both x(t ) and y (t ) are differentiable and there is no time t at which both
x(t ) and y(t ) are simultaneously 0.
(2) Let ‘s check whether the formula you derived in (1) is reasonable. For each of the
following curves, i. sketch the find a simple parameterization, ii. find the length using
the formula you found in (1), iii. Find the fomula using elementary geometry to check
the answer from ii.
a. The straight line segment from (0,0) to (1,0).
b. The straight line segment from (0,0) to (2,3).
IMSA BC 2-3
Arc Length p.1
Fall 12
(2) continued
c. The circle of radius 3 centered at the origin.
For (3) and (4) below, sketch the graph of each curve. Then set up integrals to find the
length of the curve and approximate each integral on your calculator or computer.
(2)
x = 2 cos(t) + 1, y = 4 sin(3t) for 0 ≤ t ≤ 2.
(3)
x = t2, y = cos(t) for 0 ≤ t ≤ 10.
IMSA BC 2-3
Arc Length p.2
Fall 12
On problems (4) and (5), find each arc length by doing the calculus and the algebra.
(4)
Let y  cosh x . Find the length of the curve to find the arc length for x [1, 4].
(5)
Let x = a cos3, y = a sin3, where a is a positive constant, for
0 ≤  ≤ 2Sketch the curve. Then find the length of the curve.
(Use symmetry.)
IMSA BC 2-3
Arc Length p.3
Fall 12