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CLP-4 Vector Calculus

Section 1.6 Integrating Along a Curve

Suppose that we have a curve \(\cC\) that is parametrized as \(\vr(t)\) with \(a\le t\le b\text{.}\) Suppose further that \(\cC\) is actually a piece of wire and that the density (i.e. mass per unit length) of the wire at the point \(\vr\) is \(\rho(\vr)\text{.}\) How do we figure out the mass of \(\cC\text{?}\) Of course we use the standard Calculus divide and conquer strategy. We select a natural number \(n\) and
  • divide the interval \(a\le t\le b\) into \(n\) equal subintervals, each of length \(\De t=\frac{b-a}{n}\text{.}\) We denote by \(t_\ell = a + \ell\De t\) the right hand end of interval number \(\ell\text{.}\)
  • Then we approximate the length of the part of the curve between \(\vr\big(t_{\ell-1}\big)\) and \(\vr\big(t_\ell\big)\) by \(\big|\vr\big(t_\ell\big)-\vr\big(t_{\ell-1}\big)\big|\) and the mass of the part of the curve between \(\vr\big(t_{\ell-1}\big)\) and \(\vr\big(t_\ell\big)\) by \(\rho\big(\vr(t_\ell)\big) \big|\vr\big(t_\ell\big)-\vr\big(t_{\ell-1}\big)\big|\text{.}\)
  • This gives us, as an approximate mass for \(\cC\) of
    \begin{equation*} \sum_{\ell=1}^n \rho\big(\vr(t_\ell)\big) \big|\vr\big(t_\ell\big)-\vr\big(t_{\ell-1}\big)\big| =\sum_{\ell=1}^n \rho\big(\vr(t_\ell)\big) \bigg|\frac{\vr\big(t_\ell\big)-\vr\big(t_{\ell-1}\big)} {t_\ell-t_{\ell-1}}\bigg|\De t \end{equation*}
Then we take the limit as \(n\rightarrow\infty\text{.}\) Assuming 1  that \(\vr(t)\) is continuously differentiable and that \(\rho(\vr)\) is continuous we get
\begin{equation*} \text{Mass of } \cC = \int_a^b \rho\big(\vr(t)\big) \left|\diff{\vr}{t}(t)\right|\,\dee{t} \end{equation*}
which we take to be a definition.

Definition 1.6.1.

  1. For a parametrized curve \(\big(x(t),y(t), z(t)\big)\text{,}\) \(a\le t\le b\text{,}\) in \(\bbbr^3\) that we call \(\cC\text{,}\) and for a function \(f(x,y,z)\text{,}\) we define
    \begin{equation*} \int_\cC f(x,y,z)\,\dee{s} =\int_a^b f\big(x(t), y(t) , z(t) \big)\sqrt{x'(t)^2+y'(t)^2+z'(t)^2}\ \dee{t} \end{equation*}
    In this notation the subscript \(\cC\) specifies the curve, and \(\dee{s}\) signifies arc length.
  2. For a curve \(y=f(x)\text{,}\) \(a\le x\le b\text{,}\) in \(\bbbr^2\) that we call \(C\text{,}\) and for a function \(g(x,y)\text{,}\) we define
    \begin{equation*} \int_C g(x,y)\,\dee{s} =\int_a^b g\big(x, f(x) \big)\sqrt{1+f'(x)^2}\ \dee{x} \end{equation*}
Suppose that we have a helical wire 2 
\begin{equation*} \vr(t) = \big(x(t)\,,\,y(t)\,,\,z(t)\big) =\big(a\cos t\,,\,a\sin t\,,\, bt\big)\qquad 0\le t\le 2\pi \end{equation*}
and that this wire has constant mass density \(\rho\text{.}\) Let's find the centre of mass of the wire. Recall that the centre of mass is \(\big(\bar x,\bar y,\bar z)\) with, for example, \(\bar x\) being the weighted average
\begin{equation*} \bar x = \frac{\int x\rho \dee{s}}{\int \rho\dee{s}} = \frac{\int x \dee{s}}{\int \dee{s}} \qquad\text{(since $\rho$ is constant)} \end{equation*}
of \(x\) over the wire. Similarly \(\bar y = \frac{\int y \dee{s}}{\int \dee{s}}\) and \(\bar z = \frac{\int z \dee{s}}{\int \dee{s}} \text{.}\) For the given curve
\begin{align*} \big(x(t)\,,\,y(t)\,,\,z(t)\big) &=\big(a\cos t\,,\,a\sin t\,,\, bt\big)\\ \big(x'(t)\,,\,y'(t)\,,\,z'(t)\big) &=\big(-a\sin t\,,\,a\cos t\,,\, b\big)\\ \diff{s}{t}(t) &=\sqrt{x'(t)^2+y'(t)^2+z'(t)^2}\\ &=\sqrt{a^2\sin^2t+a^2\cos^2t+b^2}\\ &=\sqrt{a^2+b^2} \end{align*}
so that
\begin{align*} \bar x &= \frac{\int x \dee{s}}{\int \dee{s}} = \frac{\int_0^{2\pi} x(t) \sqrt{a^2+b^2}\,\dee{t}} {\int_0^{2\pi} \sqrt{a^2+b^2}\,\dee{t}} = \frac{\int_0^{2\pi} a\cos(t) \,\dee{t}}{2\pi} =0\\ \bar y &= \frac{\int y \dee{s}}{\int \dee{s}} = \frac{\int_0^{2\pi} y(t) \sqrt{a^2+b^2}\,\dee{t}} {\int_0^{2\pi} \sqrt{a^2+b^2}\,\dee{t}} = \frac{\int_0^{2\pi} a\sin(t) \,\dee{t}}{2\pi} =0\\ \bar z &= \frac{\int z \dee{s}}{\int \dee{s}} = \frac{\int_0^{2\pi} z(t) \sqrt{a^2+b^2}\,\dee{t}} {\int_0^{2\pi} \sqrt{a^2+b^2}\,\dee{t}} = \frac{\int_0^{2\pi} bt \,\dee{t}}{2\pi} =\frac{b}{2\pi}\Big[\frac{t^2}{2}\Big]_0^{2\pi} =b\pi \end{align*}
So the centre of mass is right on the axis of the helix, half way up, which makes perfect sense.

Exercises Exercises

Exercise Group.

Exercises β€” Stage 1
1.
Give an equation for the arc length of a curve \(C\) as a line integral.
2.
  1. Show that the integral \(\int_\cC f(x,y)\,ds\) along the curve \(\cC\) given in polar coordinates by \(r=r(\theta)\text{,}\) \(\theta_1\le \theta\le\theta_2\text{,}\) is
    \begin{equation*} \int_{\theta_1}^{\theta_2}f\big(r(\theta)\cos\theta, r(\theta)\sin\theta\big) \sqrt{r(\theta)^2+\left(\diff{r}{\theta}(\theta)\right)^2}\, \dee{\theta} \end{equation*}
  2. Compute the arc length of \(r=1+\cos\theta,\ 0\le \theta\le 2\pi\text{.}\) You may use the formula
    \begin{equation*} 1+\cos\theta=2\cos^2\frac{\theta}{2} \end{equation*}
    to simplify the computation.

Exercise Group.

Exercises β€” Stage 2
3.
Calculate \(\int_C \left(\frac{xy}{z}\right)\dee{s}\text{,}\) where \(C\) is the curve \(\left( \frac23t^3 , \sqrt{3}t^2 , 3t \right)\) from \(t=1\) to \(t=2\text{.}\)
4.
A hoop of radius \(1\) traces out the curve \(x^2+y^2=1\text{,}\) where \(x\) and \(y\) are measured in metres. At a point \((x,y)\text{,}\) its density is \(x^2\) kg per metre. What is the mass of the hoop?
5.
Compute \(\int_C (xy+z) \dee{s}\) where \(C\) is the line segment from \((1,2,3)\) to \((2,4,5)\text{.}\)
6.
Evaluate the path integral \(\int_\cC f(x,y,z)\,\dee{s}\) for
  1. \(f(x,y,z)=x\cos z\text{,}\) \(\cC:\vr(t)=t\hi+t^2\hj\text{,}\) \(0\le t\le 1\text{.}\)
  2. \(f(x,y,z)=\frac{x+y}{y+z}\text{,}\) \(\cC:\vr(t)= \big(t,\frac{2}{3}t^{3/2},t\big)\text{,}\) \(1\le t\le 2\text{.}\)
7.
Evaluate \(\int_C \sin x\,\dee{s}\text{,}\) where \(C\) is the curve \((\arcsec(t), \ln t)\text{,}\) \(1 \le t \le \sqrt{2}\text{.}\)

Exercise Group.

Exercises β€” Stage 3
8. (✳).
Evaluate the line integral \(\int_C \vF\cdot\hn\,\dee{s}\) where \(\vF(x,y) = xy^2 \,\hi + ye^x \,\hj\) , \(C\) is the boundary of the rectangle \(R\text{:}\) \(0 \le x \le 3\text{,}\) \(-1 \le y \le 1\text{,}\) and \(\hn\) is the unit vector, normal to \(C\text{,}\) pointing to the outside of the rectangle.
9. (✳).
Let \(\cC\) be the curve given by
\begin{equation*} \vr(t)=t\cos t\,\hi+t\sin t\,\hj+t^2\,\hk,\qquad 0\le t\le \pi \end{equation*}
  1. Find the unit tangent \(\hT\) to \(\cC\) at the point \((-\pi,0,\pi^2)\text{.}\)
  2. Calculate the line integral
    \begin{equation*} \int_\cC \sqrt{x^2+y^2}\ \dee{s} \end{equation*}
  3. Find the equation of a smooth surface in \(3\)-space containing the curve \(\cC\text{.}\)
  4. Sketch the curve \(\cC\text{.}\)
10.
A wire traces out a path \(C\) described by the curve \((t+\frac12t^2 , t-\frac12t^2 , \frac{4}{3}\,t^{3/2})\text{,}\) \(0 \leq t \leq 4\text{.}\) Its density at the point \((x,y,z)\) is \(\rho(x,y,z)={\left( \frac{x+y}{2}\right)}\text{.}\) Find its centre of mass.
We could relax these conditions somewhat by instead assuming that \(\vr'(t)\) and \(\rho(t)\) are bounded and are continuous except at a finite number of points. (\(\vr'(t)\) need not exist at all at those points.)
For example, your favourite solenoid or spring or slinky.