Question

Find the volume cut from the sphere rho=7 by the cone ϕ=4. (Use symbolic notation and fractions where needed.)

Answer 1

To find the volume cut from the sphere rho=7 by the cone phi=4, we need to first find the intersection points of the sphere and the cone. Then, we can calculate the volume by finding the difference in volume between the original sphere and the spherical cap cut by the cone.

Step-by-step explanation:

To find the volume cut from the sphere rho=7 by the cone phi=4, we need to first find the intersection points of the sphere and the cone.

Then, we can calculate the volume by finding the difference in volume between the original sphere and the spherical cap cut by the cone.

1. Find the intersection points by setting rho=7 and phi=4: r=7sin(theta)cos(phi) and phi=4
2. Substitute r=7sin(theta)cos(phi) into the equation of the sphere: x^2+y^2+z^2=49
3. Calculate the limits of integration for theta and phi based on the intersection points
4. Use the formula for the volume of a spherical cap to calculate the volume of the cut portion of the sphere
5. Subtract the volume of the spherical cap from the volume of the original sphere to find the volume cut by the cone