?

Multivariable Calculus

Free Version

Upgrade subject to access all content

Difficult

Centroid of Ice Cream Cone

MVCALC-V1YNYX

Find the center of mass for a constant-density solid bounded by the sphere:

$$S_1=\{(x,y,z): x^2+y^2+z^2=a^2, z\geq 0, a \text{ is a positive constant}\}$$

...and cone:

$$S_2=\{(x,y,z): z\tan\alpha=\sqrt{x^2+y^2}, \alpha\in(0,\frac{\pi}{2})\}$$

A

$(0,0,\frac{3a}{8}(1+\cos\alpha))$

B

$(1,1,\frac{3a}{4}\sin^2\frac{\alpha}{2})$

C

$(0,0,\frac{3a}{8}\cos^2\frac{\alpha}{2})$

D

$(0,0,\frac{3a}{4}(1+\sin\alpha))$

E

$(0,0,a\cos\alpha)$