Home»Math Guides»Line Integrals (Example 1)
Examples of Line Integrals (Example 1)
You might be asked to find the line integral, given a vector valued function, a curve, and a time domain.
Example problem: Find the line integral \[\int\limits_C^{} {\mathop F\limits^ \to } \cdot dr\] given \[\mathop F\limits^ \to (x,y,z) = x\mathop i\limits^ \to + y\mathop j\limits^ \to + xy\mathop k\limits^ \to \] and \[\mathop r\limits^ \to (t) = \cos (t)\mathop i\limits^ \to + \sin (t)\mathop j\limits^ \to + t\mathop k\limits^ \to \] for \[0 \le t \le \pi \]
To evaluate the line integral, you first need to find \(F\left( {r(t)} \right) \cdot r'(t)\)
We can start by finding the derivative of r(t).
\[r'(t) = – \sin (t)\mathop i\limits^ \to + \cos (t)\mathop j\limits^ \to + \mathop k\limits^ \to \]
Then we find \[F\left( {r(t)} \right) \cdot r'(t)\]
Remember, the dot product will just be a scalar.
\[F\left( {r(t)} \right) \cdot r'(t) = \left\langle {\cos (t),\sin (t),\cos (t)\sin (t)} \right\rangle \cdot \left\langle { – \sin (t),\cos (t),1} \right\rangle \]
\[F\left( {r(t)} \right) \cdot r'(t) = – \cos (t) + \sin (t) + \cos (t)\sin (t) + \cos (t)\sin (t)\]
\[F\left( {r(t)} \right) \cdot r'(t) = \sin (t)\cos (t)\]
And now we just find the integral.
We’re told that t goes from 0 to pi, so that’s our integral boundary.
\[\int\limits_C^{} {\mathop F\limits^ to \cdot dr = } \int_0^\pi {sin (t)cos (t)dt} \]
And we use regular substitution to solve the integral.
\[u = sin (t)\]
\[du = cos (t)dt\]
\[u(0) = 0\]
\[u(\pi ) = 0\]
\[= \int_0^0 {udt = 0} \]
If you’re wondering about how we got new integral bounds that happened to go from 0 to 0, it’s basically because the integral has an equal area going above the graph and below the graph, so the net area for the integral just ends up being 0.
Alternatively, use the trigonometric identity \((sin (t)cos (t) = \frac{1}{2}sin (2t))\) to solve the integral even faster – it pays off to memorize the important trigonometric identities, but you’re not screwed if you didn’t memorize it!