### Examples of Line Integrals (Example 2)

Find the line integral $$\int\limits_C^{} {\mathop F\limits^ \to } \cdot dr$$ given $$\mathop F\limits^ \to (x,y,z) = xy\mathop i\limits^ \to + 3{y^2}\mathop j\limits^ \to$$ and $$\mathop r\limits^ \to (t) = 11{t^4}\mathop i\limits^ \to + {t^3}\mathop j\limits^ \to$$ for $$0 \le t \le 1$$.

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).

$r'(t) = 44{t^3}\mathop i\limits^ \to + 3{t^2}\cos (t)\mathop j\limits^ \to$

We also need $$F\left( {r(t)} \right)$$. F is a function of x and y, so we substitute in the i component as x and the j component as y.

$F\left( {r(t)} \right) = \left\langle {\left( {11{t^4}} \right)\left( {{t^3}} \right),3{{\left( {{t^3}} \right)}^2}} \right\rangle$

$F\left( {r(t)} \right) = \left\langle {11{t^7},3{t^6}} \right\rangle$

Then we find $$F\left( {r(t)} \right) \cdot r'(t)$$. The dot product is a scalar, but in this case we can't combine variables of different powers so there are multiple terms.

$F\left( {r(t)} \right) \cdot r'(t) = \left\langle {11{t^7},3{t^6}} \right\rangle \cdot \left\langle {44{t^3},3{t^2}} \right\rangle$

$F\left( {r(t)} \right) \cdot r'(t) = 484{t^{10}} + 9{t^8}$

And now we just find the integral. We're told that t goes from 0 to 1, so that's our integral boundary.

$\int\limits_C^{} {\mathop F\limits^ \to \cdot dr = } \int_0^1 {\left( {484{t^{10}} + 9{t^8}} \right)dt}$

$\int\limits_C^{} {\mathop F\limits^ \to \cdot dr = } \frac{{484}}{{11}} + \frac{9}{9}$

$\int\limits_C^{} {\mathop F\limits^ \to \cdot dr = } 45$

Please see our other worked-out line integral examples:

Line Integrals (Ex1)

Line Integrals Multiple Line Segments (Ex3)

Line Integrals Parametrizing Curves (Ex4)