find out the volume of solid removed?

Question

i have sphere that has an equation $$x^2+y^2+z^2=1$$ a cylindrical hole $x^2+(y-1/2)^2$=$1/4$ is cut through it . find the volume of the portion cut. i don't know what to do, i was thinking of using cavalieri's principle , and integrate with respect to $dz$ but i could not imagine the equation of sheets that would be formed . i guess we can proceed with polar , but i don't know how? please help

Answer 1

To integrate in yucky cartesian coordinates you would do for the sphere:

$$V_\textrm{sphere} =\int_{-1}^1dx\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}dy\int_{-\sqrt{1-y^2-x^2}}^{\sqrt{1-y^2-x^2}}dz = 8*\int_0^1dx\int_0^{\sqrt{1-x^2}}dy\int_0^{\sqrt{1-y^2-x^2}}dz$$

(to see this worked out in some detail see: https://www.youtube.com/watch?v=pIf78H50iNc)

To get the missing cylinder, you need to integrate over the region of the cylinder, Same $z$ bounds, but just integrate $x$ and $y$ over the cylindrical region given by your equation.

viz. $$\int_{-1/2}^{1/2}dx\int_{(1/2) -\sqrt{1/4-x^2}}^{(1/2) +\sqrt{1/4-x^2}}dy\int_{-\sqrt{1-y^2-x^2}}^{\sqrt{1-y^2-x^2}}dz $$