On an Exam, Students Were Asked to Evaluate $\int _ { C } \left( x ^ { 2 } \vec { i } + x \vec { j } \right) \cdot d \vec { r }$

question 44

Essay

On an exam, students were asked to evaluate $\int _ { C } \left( x ^ { 2 } \vec { i } + x \vec { j } \right) \cdot d \vec { r }$ , where C has the parameterization $x = \cos t , y = \sin t , 0 \leq t \leq \pi$ .One student wrote:
"Using Green's Theorem, $\int _ { C } \left( x ^ { 2 } \vec { i } + x \vec { j } \right) \cdot d \vec { r } = \int _ { D } \left( \frac { \partial x } { \partial x } - \frac { \partial \left( x ^ { 2 } \right) } { \partial y } \right) d A = \int _ { D } 1 d A =$ Area of the semi-circle = $\frac { \pi } { 2 }$ ."
Do you agree with the student?

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