Arc Length problem
Use Riemann sum and integrals:
Area (approximation: area of rectangle)
Volume (approximation: volume of cylinder)
Arc Length (approximation: length of straight line)
Arc Length problem: consider a function y = f (x), a ≤ x ≤ b, what is the length of
the graph of y = f (x) from the point (a, f (a)) to (b, f (b))? (Think if you have a soft
ruler, so you can measure it)
Distance between two points (x1 , y1 ) and (x2 , y2 ):
q
d = (x1 − x2 )2 + (y1 − y2 )2
Derivation of the formula
Divide the interval [a, b] to n equal subintervals, [x0 , x1 ], [x1 , x2 ], · · · , [xn−1 , xn ], each
with length ∆x = (b − a)/n, and x0 = a, · · · , xi = xi−1 + ∆x, · · · , xn = b.
We approximate the arc length between (xi−1 , f q
(xi−1 )) and (xi , f (xi )) by a straight
line segment. The length of the line segment is (xi − xi−1 )2 + [f (xi ) − f (xi−1 )]2 .
0
By using mean value theorem (see Chapter 4), f (xi ) − f (xi−1
q ) = f (yi )(xi − xi−1 ),
where yi is a point in (xi−1 , xi ). Thus the distance is now 1 + [f 0 (yi )]2 ∆x. So we
get a Riemann sum by summing the approximate arc length:
d=
n q
X
1 + [f 0 (yi )]2 ∆x,
xi−1 < yi < xi .
i=1
Arc Length Formula: consider a function y = f (x), a ≤ x ≤ b, the length of the
graph of y = f (x) from the point (a, f (a)) to (b, f (b)) is
b
Z
L=
a
q
1 + [f 0 (x)]2 dx.
Examples
Example 1: Find the length of the curve y =
(calculate the arc length of a half-circle)
√
r 2 − x 2 from (−r , 0) to (r , 0).
Example 2: Find the length of the curve y = x 2 from (0, 0) to (1, 1).
Example 3: Find the length of the curve y =
x3
1 1
+
, ≤ x ≤ 1.
6
2x 2
Example 4: Find the length of the curve y = ln(cos x), 0 ≤ x ≤ π/3.
Most integrals for arc length are hard to solve because of radicals, so one may set up
the integral and use approximation methods (like, trapezoid, Simpson rules) to solve it
numerically.
Work (an application of integral to physics)
Work measures the amount of effort required to perform a task.
Basic law: Work = force × distance (W = F · d)
(when the force is a constant)
Example 1: (a) You carry a 10 kilogram luggage for 200 meter. How much work is
done? (b) You lift a 2-lb book off the table to 1 foot high. How much work is done?
Unit: (a) joule=Newton × meter (SI metric); (b) foot × lb (US)
(9.8 Newton= 1 kg ×g )
Distance and velocity: Distance= velocity × time (constant v )
Z b
D=
v (t)dt (variable velocity v (t))
a
Work and force: Work = force × distance (constant F )
Z b
F (x)dx (variable force F (t))
W =
a
Mass and density: Mass =density × volume (constant D)
Z b
M=
D(x)dx (variable density D(x))
a
From multiplication to integral
Work and force: Work = force × distance (constant F )
Z b
W =
F (x)dx (variable force F (t))
a
Example 2: A particle is moved along the x-axis by a force that measures 10/(1 + x)2
lb at a point x ft from the origin. Find the work done in moving the particle from the
origin to a distance of 9 ft.
Example 3: (spring problem) A spring has a natural length of 20 cm. If a 25-N force is
required to keep it stretched to a length of 30 cm, how much work is required to
stretch it from 20 cm to 25 cm ?
Hooke’s law: (pulling force) f (x) = kx, where x is the stretched length beyond the
natural length, and k is the spring constant.
Example 4: (pulling cable (rope) problem) A 200-lb cable is 100 ft long and hang
vertically from the top of a tall building. How much work is required to lift the cable
to the top of the building?
Difficulty: no direct formula for f (x).
Use Riemann sum for work
Step 1: Take a small piece with thickness ∆x on the object where work is done.
Step 2: Approximate the work required for the small piece.
Step 3: Form a Riemann sum from Step 2, and take limit to obtain a definite integral
to represent the work.
Step 4: Solve the integral.
Example 5: (pumping water problem) A circular swimming pool has a diameter of 24
ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is
required to pump all of the water out over the side? (Use the fact that water weight
62.5 lb/ft3 ) What if the pool is a inverted circular cone shaped?