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CSS Applied Mathematics QUESTION #4165
Question 1
Using Green's theorem, $\oint_C \vec{F} \cdot d\vec{r} = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$ where $\vec{F} = [\frac{1}{2}xy^4, \frac{1}{2}x^4y]$. For the rectangle $[0,3] \times [0,2]$, what is the value?
  • $6$
  • $12$
  • $18$✔️
  • $36$
Correct Answer Explanation
$\frac{\partial Q}{\partial x} = 2x^3y$, $\frac{\partial P}{\partial y} = 2xy^3$. So $\iint_R (2x^3y - 2xy^3) dA = \int_0^3\int_0^2 (2x^3y - 2xy^3) dy dx = 18$.