Answer
(a) $\mathop \smallint \limits_{ - {C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - 8$
(b) $\mathop \smallint \limits_{{C_2}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - 11$
(c) $\mathop \smallint \limits_{ - {C_1} - {C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - 16$
Work Step by Step
(a) Write
$\mathop \smallint \limits_{ - {C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - \mathop \smallint \limits_{{C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}}$
Since $\mathop \smallint \limits_{{C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = 8$, so $\mathop \smallint \limits_{ - {C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - 8$.
(b) Write
$\mathop \smallint \limits_C^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = 5 = \mathop \smallint \limits_{{C_1}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} + \mathop \smallint \limits_{{C_2}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} + \mathop \smallint \limits_{{C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}}$
Since $\mathop \smallint \limits_{{C_1}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = 8$, $\mathop \smallint \limits_{{C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = 8$, so
$5 = 8 + \mathop \smallint \limits_{{C_2}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} + 8$
$\mathop \smallint \limits_{{C_2}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - 11$
(c) Write
$\mathop \smallint \limits_{ - {C_1} - {C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = \mathop \smallint \limits_{ - \left( {{C_1} + {C_3}} \right)}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - \mathop \smallint \limits_{{C_1} + {C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - \left( {\mathop \smallint \limits_{{C_1}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} + \mathop \smallint \limits_{{C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}}} \right)$
Since $\mathop \smallint \limits_{{C_1}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = 8$, $\mathop \smallint \limits_{{C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = 8$, so
$\mathop \smallint \limits_{ - {C_1} - {C_3}}^{} {\bf{F}}\cdot{\rm{d}}{\bf{r}} = - 16$
