- #1
mathusers
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hi how can the following be proved using integral methods:
a) prove surface area of sphere, radius a, is 4 [tex]\pi a^2[/tex]
b) prove area of a disk, radius a, is [tex]\pi a^2[/tex]
c) prove volume of ball, radius a, is [tex]\frac{4}{3} \pi a^3[/tex]
d) prove volume of axisymmetric cone of height h and base with radius a, is [tex]\frac{1}{3}\pi a^2 h[/tex]
......... ...
i think my working of (a) is correct:
working of (a):
use spherical co-ordinates:
|S| = [tex]\int\int_{D} ||\frac{dr}{d\theta} X \frac{dr}{d\phi}|| dA[/tex]
[tex]||\frac{dr}{d\theta} X \frac{dr}{d\phi}|| = a^2sin \phi[/tex]
so:
[tex]|S| = \int^{2\pi}_{0}\int^{\pi}_{0} a^2sin \phi d\phi d\theta = \int^{2\pi}_{0} \left[ -a^2cos\phi \right]^{\pi}_0 d\theta = \int^{2\pi}_{0} 2a^2 d\theta = 4\pi a^2 [/tex]
.........
how can i do the rest please. and what integration methods should I be using for each? thnx xxxx
a) prove surface area of sphere, radius a, is 4 [tex]\pi a^2[/tex]
b) prove area of a disk, radius a, is [tex]\pi a^2[/tex]
c) prove volume of ball, radius a, is [tex]\frac{4}{3} \pi a^3[/tex]
d) prove volume of axisymmetric cone of height h and base with radius a, is [tex]\frac{1}{3}\pi a^2 h[/tex]
......... ...
i think my working of (a) is correct:
working of (a):
use spherical co-ordinates:
|S| = [tex]\int\int_{D} ||\frac{dr}{d\theta} X \frac{dr}{d\phi}|| dA[/tex]
[tex]||\frac{dr}{d\theta} X \frac{dr}{d\phi}|| = a^2sin \phi[/tex]
so:
[tex]|S| = \int^{2\pi}_{0}\int^{\pi}_{0} a^2sin \phi d\phi d\theta = \int^{2\pi}_{0} \left[ -a^2cos\phi \right]^{\pi}_0 d\theta = \int^{2\pi}_{0} 2a^2 d\theta = 4\pi a^2 [/tex]
.........
how can i do the rest please. and what integration methods should I be using for each? thnx xxxx