Math 241: Problem of the day
Problem: Find the velocity, speed, and acceleration of the curve
r(t) = ht 2 , t 3 i for t ∈ [1, 3]. Calculate the length of the curve.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Integrating functions over curves.
C a curve.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Integrating functions over curves.
C a curve. Parameterization: r(t) = hx(t), y (t)i, t ∈ [a, b],
continuously differentiable.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Integrating functions over curves.
C a curve. Parameterization: r(t) = hx(t), y (t)i, t ∈ [a, b],
continuously differentiable.
f : R2 → R a function.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Integrating functions over curves.
C a curve. Parameterization: r(t) = hx(t), y (t)i, t ∈ [a, b],
continuously differentiable.
f : R2 → R a function.
Define the integral of f over C (against arc length)
Z
Z
f ds =
C
b
f (r(t)) kr0 (t)k dt.
a
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Integrating functions over curves.
C a curve. Parameterization: r(t) = hx(t), y (t)i, t ∈ [a, b],
continuously differentiable.
f : R2 → R a function.
Define the integral of f over C (against arc length)
Z
Z
b
f ds =
C
“ds” is arc length =
f (r(t)) kr0 (t)k dt.
a
kr0 (t)kdt.
Z
⇒
ds = length of C .
C
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Integrating functions over curves.
C a curve. Parameterization: r(t) = hx(t), y (t)i, t ∈ [a, b],
continuously differentiable.
f : R2 → R a function.
Define the integral of f over C (against arc length)
Z
Z
b
f ds =
C
“ds” is arc length =
f (r(t)) kr0 (t)k dt.
a
kr0 (t)kdt.
Z
⇒
ds = length of C .
C
Meaning?
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Average value.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Average value.
Average value of f over the curve C :
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Average value.
Average value of f over the curve C :
1
Avg(f , C ) =
length(C )
Z
f ds.
C
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Average value.
Average value of f over the curve C :
1
Avg(f , C ) =
length(C )
Z
f ds.
C
Example. f (x, y ) = distance to y –axis, C = circle of radius 1
centered at (1, 0). Calculate average distance.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Density to mass.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Density to mass.
A piece of wire in shape of a curve C
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Density to mass.
A piece of wire in shape of a curve C
f (x, y ) = density at point (x, y ) of wire.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Density to mass.
A piece of wire in shape of a curve C
f (x, y ) = density at point (x, y ) of wire.
Mass M =
R
C
fds.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Density to mass.
A piece of wire in shape of a curve C
f (x, y ) = density at point (x, y ) of wire.
Mass M =
R
C
fds.
Center of mass (¯
x , y¯ ):
Z
1
x¯ =
x f (x, y ) ds
M C
1
y¯ =
M
Z
y f (x, y ) ds.
C
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Area between curve and graph.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Area between curve and graph.
R
Can also interpret C f ds as the signed area of the surface
between C and the graph.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Area between curve and graph.
R
Can also interpret C f ds as the signed area of the surface
between C and the graph.
Example. Find area of cylinder x 2 + y 2 = 1 between xy –axis and
z = 2 + x 2.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Other facts.
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Other facts.
• All of this works for curves in Rn .
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html
Other facts.
• All of this works for curves in Rn .
• “Independent of the parameterization”
www.math.uiuc.edu/∼fcellaro/Francesco Cellarosi Home Page/241.html