Solve Volume of Sphere & Cone w/ Cylinder | Urgent

[yongkiat]

[URGENT]ow to solve this?

Do i need to use any theorem to solve this such as divergence theorem?

the first question need to find the volume of the sphere that a part in up and bottom being cut( z-axis) and the middle is a hole( cylindrical shape )

the second question need to find the volume of a cone with a cylinder.

for both question, the answer is not need completely solve, just need find the main equation for finding the volume.( I mean just find the range for integration by using dx,dy,dz or using polar coordinate )
Equation for finding volume equation using polar coordinate is prefered.

 

[zhentil]

Why would divergence theorem be needed? For the first one, z = +- sqrt(j^2-x^2-y^2). Put this in a polar integral.

 

[tacman]

To solve for the volume of a sphere with a cut in the middle, you can use the integral form of the volume formula for a sphere, which is V = (4/3)πr^3. However, since there is a cylindrical hole in the middle, you will need to subtract the volume of the cylinder from the total volume of the sphere. The equation for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height of the cylinder. So, the final equation for the volume of the sphere with a cut and a cylindrical hole would be V = (4/3)πr^3 - πr^2h, where r is the radius of the sphere and h is the height of the cylinder.

To solve for the volume of a cone with a cylinder, you can use the same approach. The volume of a cone is V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone. So, the final equation for the volume of a cone with a cylinder would be V = (1/3)πr^2h + πr^2h, where r is the radius of the base and h is the height of the cylinder.

As for using the divergence theorem, it is not necessary for these problems as they can be solved using basic geometry and volume formulas. However, if you are familiar with the divergence theorem and would like to use it, you can apply it to find the volume of a sphere or cone with a cylindrical hole. The divergence theorem states that the volume of a solid can be found by integrating the divergence of its vector field over its boundary. This may be a more complex approach, but it can also be used to solve these types of problems.