Preview Activity 13.3.1.
Let \(\vF=\langle xy,y^2\rangle\text{,}\) let \(C_1\) be the line segment from \((1,1)\) to \((4,1)\text{,}\) let \(C_2\) be the line segment from \((4,1)\) to \((4,3)\text{,}\) and let \(C_3\) be the line segment from \((1,1)\) to \((4,3)\text{.}\) Also let \(C = C_1 + C_2\text{.}\) This vector field and the curves are shown in Figure 13.3.1.
The vector field \(\vF=\langle xy,y^2\rangle\) plotted for \(0\leq x\leq 5\) and \(0\leq y\leq 4\text{.}\) This vector field radiates from the origin with vector length increasing as distance from the origin increases. The graph includes the line segment \(C_1\) from \((1,1)\) to \((4,1)\text{,}\) the line segment \(C_2\) from \((4,1)\) to \((4,3)\text{,}\) and the line segment \(C_3\) from \((1,1)\) to \((4,3)\text{.}\)
(a)
Every point along \(C_1\) has \(y=1\text{.}\) Therefore, along \(C_1\text{,}\) the vector field \(\vF\) can be viewed purely as a function of \(x\text{.}\) In particular, along \(C_1\text{,}\) we have \(\vF(x,1) = \langle x,1\rangle\text{.}\) Since every point along \(C_2\) has the same \(x\)-value, write \(\vF\) as a function of \(y\) only (for the points on \(C_2\)).
(b)
Recall that \(d\vr \approx \Delta \vr\text{,}\) and along \(C_1\text{,}\) we have that \(\Delta\vr = \Delta x\vi \approx dx\vi\text{.}\) Thus, \(d\vr = \langle dx,0\rangle\text{.}\) We know that along \(C_1\text{,}\) \(\vF = \langle x,1\rangle\text{.}\)
(i)
(ii)
(iii)
Write \(\int_{C_1} \vF\cdot d\vr\) as an integral of the form \(\int_a^b f(x)\, dx\) and evaluate the integral.
(c)
Use an analogous approach to write \(\int_{C_2} \vF\cdot d\vr\) as an integral of the form \(\int_c^d g(y)\, dy\) and evaluate the integral.
(d)
Use the previous parts and a property of line integrals to calculate \(\int_C\vF\cdot d\vr\) without having to evaluate any additional integrals.
