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As the first bullet point tell us, $ \begin{eqnarray*} d\vec{A} &=& R^2sin\theta d\theta d\phi \hat{r} \\ \implies \vec A &=& \int \int R^2 sin\theta \hat{r}d\theta d\phi \\ \vec{A} &=& \int_{0}^{2\pi} \int_{0}^{1/2} R^2 sin\theta \hat{r}d\theta d\phi \\ \end{eqnarray*} $ Sub $\hat{r} = sin\theta cos\phi \hat{x} + sin \theta sin\phi \hat{y} + cos\theta \hat{z} $ into integral to give, \begin{eqnarray*} \vec{A} &=& \hat{x}(R^2 \int_{0}^{2\pi} cos \phi d\phi \int_{0}^{\frac{1}{2}} sin^{2} \phi d\phi) + \hat{y}(R^{2}\int_{0}^{2 \pi} sin\phi d\phi \int_{0}^{\frac{1}{2}} sin^{2}\theta d\theta)+\\ \hat{z}(R^{2} \int_{0}^{2\pi} d\phi\int_{0}^{\frac{1}{2}}sin\theta cos\theta d\theta)\\ &=& \hat{x}(0) + \hat{y}(0) + \hat{z}(R^{2} \int_{0}^{2\pi} d\phi\int_{0}^{\frac{1}{2}}sin\theta cos\theta d\theta)\\ &=& \hat{z} (R^2 (2\pi)(0.1149)) \end{eqnarray*} We are told R= 6371 km, hence, $\vec{A} = \hat{z}(29309373km^2) $
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Wed, 22 Apr 2020 22:24 GMT