Question Number 227341 by fantastic2 last updated on 17/Jan/26
$${find}\:{the}\:{surface}\:{area}\:{of}\:{a}\:{spherical} \\ $$$${cap} \\ $$
Commented by Kassista last updated on 17/Jan/26
$$ \\ $$$${are}\:{you}\:{simply}\:{asking}\:{to}\:{prove}\:{the}\:{surface}\:{area} \\ $$$$\:{of}\:{the}\:{sphere}? \\ $$
Commented by fantastic2 last updated on 17/Jan/26
$${no} \\ $$
Commented by fantastic2 last updated on 17/Jan/26
Commented by fantastic2 last updated on 17/Jan/26
$${find}\:{surface}\:{area}\:{of}\:{blue}\:{region} \\ $$
Commented by mr W last updated on 17/Jan/26
$${A}=\mathrm{2}\pi{rh}+\pi{a}^{\mathrm{2}} \\ $$
Commented by fantastic2 last updated on 17/Jan/26
$${how} \\ $$$$ \\ $$
Commented by mr W last updated on 17/Jan/26
$${A}_{\mathrm{1}} =\int_{\mathrm{0}} ^{\theta} \mathrm{2}\pi{r}\:\mathrm{sin}\:\alpha\:{rd}\alpha \\ $$$$\:\:=\mathrm{2}\pi{r}^{\mathrm{2}} \int_{\mathrm{0}} ^{\theta} \mathrm{sin}\:\alpha\:{d}\alpha \\ $$$$\:\:=\mathrm{2}\pi{r}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{cos}\:\theta\right) \\ $$$$\:\:=\mathrm{2}\pi{rh} \\ $$$${A}_{\mathrm{2}} =\pi{a}^{\mathrm{2}} \\ $$
Commented by fantastic2 last updated on 17/Jan/26
$${great} \\ $$