How to find the volume of the unit sphere above $z=1/\sqrt2$?

I have managed to find the limit for $\rho$ as $(1/\sqrt2) \sec \phi$ to $1$ and for $\theta$ as $0$ to $2\pi$

but I can't figure out the limits for $\phi$.

Can anyone help?

This is the problem by the way: http://ocw.covenantuniversity.edu.ng/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/4.-triple-integrals-and-surface-integrals-in-3-space/part-a-triple-integrals/session-77-triple-integrals-in-spherical-coordinates/MIT18_02SC_L26Brds_10.png

Thanks.

In spherical coordinates, the region can be written as $$ E=\{(\rho,\theta,\phi)\;|\; 0 \le \theta \le 2\pi , \frac{1}{\sqrt{2}\cos\phi}\le \rho \le 1,0 \le \phi\le \cos^{-1}\left( \frac{1}{\sqrt{2}}\right)(=\frac{\pi}{4})\} $$ It follows that the volume equals $$ \iiint_E \rho^2\sin \phi\; dV = \frac{2\pi}{3}-\frac{\pi}{\sqrt{2}}+\frac{\pi}{6\sqrt{2}} $$

Note.

  • To specifically answer your question, the upper bound for $\phi$ comes from equations $$ z=\frac{1}{\sqrt{2}}=\rho \cos \phi, \quad \rho=1 $$
  • Cylindrical coordinates are also a good option here: $$ E=\{(r,\theta,z)\;|\; 0 \le \theta \le 2\pi , 0\le r\le \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \le z \le \sqrt{1-r^2} \} $$

With a generic sphere of radius R,if you slice perpendicular to the z axis the slices are circles of radius $\sqrt{R^2-z^2}$ and the area of each slice is $\pi(R^2-z^2)$

Volume of a sphere = $\pi\int_{-R}^R (R^2 - z^2) dz$

The cap of the unit sphere above $z = \dfrac{1}{\sqrt2}$

$\pi \int_{\frac{1}{\sqrt2}}^1 (1 - z^2) dz$

So, you still want to do a triple integral...

$\int_0^{2\pi}\int_0^{\pi/4}\int_{\frac{\sec\phi}{\sqrt2}}^{1} \rho^2 \sin\phi\, d\rho\, d\phi\, d\theta$

$\int_0^{2\pi}\int_0^{\pi/4}\frac{1}{3}\rho^3 \sin\phi\, d\phi\, d\theta|_{\frac{\sec\phi}{\sqrt2}}^{1}$

$\int_0^{2\pi}\int_0^{\pi/4}\frac{1}{3}\sin\phi\ (1-\frac{1}{2^{3/2}\cos^3\phi})\, d\phi\, d\theta$

$\int_0^{2\pi}\int_0^{\pi/4}\frac{1}{3}\sin\phi\ d\phi\, d\theta$ -$\int_0^{2\pi}\int_0^{\pi/4}\dfrac{\sin\phi}{2^{3/2}\cos^3\phi}\, d\phi\, d\theta$

and each of those should be pretty straightforward.

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.