Examples of Line Integrals (Example 1)

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.

\[u = \sin (t)\]

\[du = \cos (t)dt\]

\[u(0) = 0\]

\[u(\pi ) = 0\]

\[\int\limits_C^{} {\mathop F\limits^ \to \cdot dr = } \int_0^0 {udt = 0} \]

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!

Please see our other worked-out line integral examples:

Line Integrals (Ex2)

Line Integrals Multiple Line Segments (Ex3)

Line Integrals Parametrizing Curves (Ex4)

Alternatively return to the line integral example hub page:

Line Integrals

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