Find the volume of the solid that lies within the sphere x^2+y^2+z^2= 36. above the xy-plane, and below the cone z=x^2+y^2 using spherical coordinates. Draw a picture.

Answers

Answer 1

The volume of the solid that lies within the sphere x^2+y^2+z^2= 36, above the xy-plane, and below the cone z=x^2+y^2 is 96π cubic units. The calculation was done using spherical coordinates.

To find the volume of the solid that lies within the sphere x^2+y^2+z^2= 36, above the xy-plane, and below the cone z=x^2+y^2, we can use spherical coordinates.

The sphere has radius 6, so we have:

0 ≤ ρ ≤ 6

The cone has equation z = ρ^2, so we have:

ρ cos(φ) = ρ^2 sin(φ)

cos(φ) = ρ sin(φ)

tan(φ) = 1/ρ

φ = π/4

Therefore, we have:

π/4 ≤ φ ≤ π/2

0 ≤ θ ≤ 2π

Using the formula for the volume element in spherical coordinates, we have:

dV = ρ^2 sin(φ) dρ dφ dθ

Integrating over the given limits, we get:

V = ∫(θ=0 to 2π) ∫(φ=π/4 to π/2) ∫(ρ=0 to 6) ρ^2 sin(φ) dρ dφ dθ

V = ∫(θ=0 to 2π) ∫(φ=π/4 to π/2) [ρ^3 sin(φ) / 3] |_ρ=0 to 6 dφ dθ

V = ∫(θ=0 to 2π) ∫(φ=π/4 to π/2) 72 sin(φ) / 3 dφ dθ

V = ∫(θ=0 to 2π) [72 cos(φ)]|φ=π/4 to π/2 dθ

V = ∫(θ=0 to 2π) 48 dθ

V = 96π

Therefore, the volume of the solid is 96π cubic units.

The solid is a spherical cap above the xy-plane and below the cone z=x^2+y^2.

picture:

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Related Questions

2.1 Convert the following: 1. 10g to Kg. 2. 32km to meter. 3. 12 m² to mm²
4. 50000mm³ to m³
5. 2,36hrs to hrs, minutes and seconds
2.2 The distance between town A and town B is 16500m. What is the distance exactly halfway between the towns in Km?

Answers

10g is equal to 0.01 Kg.

32km is equal to 32,000 meters.

12 m² is equal to 12,000 mm².

50,000mm³ is equal to 0.05 m³.

2.36hrs is equal to 2 hours, 21 minutes, and 36 seconds.

The distance exactly halfway between town A and town B is 8.25 km.

To convert grams to kilograms, divide the given value by 1000 since there are 1000 grams in a kilogram.

To convert kilometers to meters, multiply the given value by 1000 since there are 1000 meters in a kilometer.

To convert square meters to square millimeters, multiply the given value by 1,000,000 since there are 1,000,000 square millimeters in a square meter.

To convert cubic millimeters to cubic meters, divide the given value by 1,000,000,000 since there are 1,000,000,000 cubic millimeters in a cubic meter.

To convert hours to hours, minutes, and seconds, the given value can be expressed as 2 hours and 0.36 hours. The decimal part represents the minutes and seconds. Multiply 0.36 by 60 to get 21.6 minutes, and then convert 0.6 minutes to seconds, which is 36 seconds.

For the second part of the question, to find the distance exactly halfway between town A and town B, divide the total distance (16500m) by 2 to get 8250m. Since the answer should be in kilometers, divide 8250 by 1000 to get 8.25 Km.

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