Use polar coordinates to find the volume of the given solid. Above the cone z = sqrt x2 + y2 and below the sphere x2 + y2 + z2 = 49?
- PopeLv 74 months agoFavorite Answer
You really, really need to work on your notation. This time I am going to help. I think these must be the two surfaces you intended:
z = √(x² + y²)
x² + y² + z² = 49
Some indication of exponent is necessary for those 2s, and in order to describe a cone, that square root argument must include both terms. Use brackets.
Both surfaces have symmetry on the z-axis. The radius of any level section of the conic surface is equal to z. The sphere has radius 7. The region you defined (sort of) is a conic sector.
Find the height of the circle where cone meets sphere.
z = r
z = √(49 - z²)
z = 7/√(2)
Now I am looking at the spherical surface of the sector.
Its height is 7 - 7/√(2) = [14 - 7√(2)]/2.
Project that surface horizontally away from the z-axis, onto the vertical cylinder circumscribing the sphere. Its image is a right cylinder having these dimensions:
radius = 7
height = [14 - 7√(2)]/2
area = 14π[14 - 7√(2)]/2 = 49[2 - √(2)]π
That also is the area of the spherical surface of the sector. Volume of a sphere sector is proportional to the area of its spherical surface.
(sector volume) : (sphere volume) = 49[2 - √(2)]π : (sphere surface area)
(sector volume) : (4/3)(7³)π = 49[2 - √(2)]π : 4(7²)π
sector volume = 343[2 - √(2)]π/3