Question

Find the vector area of a hemispherical bowl of radius r.

Answer 1

Solution :- To find the vector area of hemispherical bowl of radius r, the vector area is given by the integral

`\vec{a}=\int_(s) d\vec{a}` for a surface S

The area for a hemisphere is given by

`d\vec{a}=R^(2) \sin\theta d\theta d \phi \hat{r}`

`\vec{a} = R^(2)\int_(0)^(\pi /2) sin\theta d\theta \cdot \int_(0)^(2\pi)d\phi \hat{r}`

by using}

`\hat{r} = sin\theta cos \phi \hat{i}+sin\theta sin\phi \hat{j}+cos\theta \hat{k}`

`\text{Now we have }\\ \vec{a} = R^(2)\int_(0)^(\pi /2) sin\theta d\theta \int_(0)^(2\pi)d\phi [sin\theta cos\phi\hat{i}+sin\theta sin\phi \hat{j}+cos\theta \hat{k} ]`

`=\hat{i}R^(2)\int_(0)^(\pi/2)sin^(2)\theta d\phi \int_(0)^(2\pi ) cos\phi d\phi + \hat{j}R^(2)\int_(0)^(\pi /2)sin\theta d\theta \int_(0)^(2 \Pi ) sin\phi d\phi+\hat{k}R^(2)\int_(0)^(\pi/2)sin{\theta}cos{\theta}d{\theta}\int_(0)^(2\pi)d\phi`

`=0+0+2{\pi}R^2\hat{k}\int_(0)^(1)(-1)tdt\\ \text {[here} \cos\theta=t\\-sin\theta d\theta=dt]\\= \pi R^2 \hat{k}`