Definition

Suppose that we can describe a curve \(C\) by the vector function \(\mathbf{r} = \mathbf{r}\left( s \right),\) \(0 \le s \le S,\) where the variable \(s\) is the arc length of the curve (Figure \(1\text{).}\)

If a scalar function \(F\) is defined over the curve \(C,\) then the integral \(\int\limits_0^S {F\left( {\mathbf{r}\left( s \right)} \right)ds} \) is called a line integral of scalar function \(F\) along the curve \(C\) and denoted as

The line integral \(\int\limits_C {Fds}\) exists if the function \(F\) is continuous on the curve \(C.\)

Properties of Line Integrals of Scalar Functions

The line integral of a scalar function has the following properties:

1. The line integral of a scalar function over the smooth curve \(C\) does not depend on the orientation of the curve;
2. If \({C_1}\) is a curve that begins at \(A\) and ends at \(B,\) and if \({C_2}\) is a curve that begins at \(B\) and ends at \(D\) (Figure \(2\)), then their union is defined to be the curve \({C_1} \cup {C_2}\) that progresses along the curve \({C_1}\) from \(A\) to \(B,\) and then along \({C_2}\) from \(B\) to \(D,\) so that
   \[\int\limits_{{C_1} \cup {C_2}} {Fds} = \int\limits_{{C_1}} {Fds} + \int\limits_{{C_2}} {Fds} ;\]



3. If the smooth curve \(C\) is parameterized by \(\mathbf{r} = \mathbf{r}\left( t \right),\) \(\alpha \le t \le \beta \) and the scalar function \(F\) is continuous on the curve \(C,\) then
   \[\int\limits_C {F\left( {x,y,z} \right)ds} = \int\limits_\alpha ^\beta {F\left( {x\left( t \right),y\left( t \right),z\left( t \right)} \right) \sqrt {{{\left( {x'\left( t \right)} \right)}^2} + {{\left( {y'\left( t \right)} \right)}^2} + {{\left( {z'\left( t \right)} \right)}^2}} dt}. \]
4. If \(C\) is a smooth curve in the \(xy\)-plane given by the equation \(y = f\left( x \right),\) \(a \le x \le b,\) then
   \[\int\limits_C {F\left( {x,y} \right)ds} = \int\limits_a^b {F\left( {x,f\left( x \right)} \right) \sqrt {1 + {{\left( {f'\left( x \right)} \right)}^2}} dx} .\]
5. Similarly, if a smooth curve \(C\) in the \(xy\)-plane is defined by the equation \(x = \varphi \left( y \right),\) \(c \le y \le d,\) then
   \[\int\limits_C {F\left( {x,y} \right)ds} = \int\limits_c^d {F\left( {\varphi \left( y \right),y} \right) \sqrt {1 + {{\left( {\varphi'\left( y \right)} \right)}^2}} dy} ;\]
6. In polar coordinates the line integral \(\int\limits_C {F\left( {x,y} \right)ds} \) becomes
   \[\int\limits_C {F\left( {x,y} \right)ds} = \int\limits_\alpha ^\beta {F\left( {r\cos \theta ,r\sin \theta } \right) \sqrt {{r^2} + {{\left( {\frac{{dr}}{{d\theta }}} \right)}^2}} d\theta } ,\]
   where the curve \(C\) is defined by the polar function \(r\left( \theta \right).\)

Solved Problems

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Solution.

Solution.

The arc length differential is

Then applying the formula

in the \(xy\)-plane, we obtain