Volume Calculation under a Paraboloid

The video tutorial explains how to calculate the volume of a solid under a paraboloid and above a circle in the XY plane using polar coordinates. It involves setting up and solving a double integral, with detailed steps on converting the function into polar form, determining integration limits, and performing the integration to find the volume.

What is the equation of the paraboloid under which the volume is to be calculated?

• Z = 16 - x^2 - y^2

• Z = x^2 + y^2 - 16

• Z = 4 - x^2 - y^2

• Z = 16 + x^2 + y^2

Why are polar coordinates used in this problem?

• Because the circle is a paraboloid

• Because the paraboloid is a cylinder

• Because the XY trace is a circle

• Because the paraboloid is a circular shape

What is the radius of the circle in the XY plane?

• 4

• 3

• 2

• 1

What is the function f(r, θ) in polar coordinates?

• 16 - r^2

• r^2 - 16

• 16 + r^2

• r^2 + 16

What are the limits of integration for R?

• 0 to 4

• 0 to 1

• 0 to 3

• 0 to 2

What are the limits of integration for Theta?

• 0 to π

• 0 to 2π

• 0 to 3π

• 0 to 4π

What is the result of integrating 16R with respect to R?

• R^2

• 4R^2

• 16R^2

• 8R^2

What is the final volume of the solid?

• 32π cubic units

• 64π cubic units

• 56π cubic units

• 28π cubic units

What is the integral of 28 with respect to Theta?

• 14θ

• 28θ

• 84θ

• 56θ

What is the value of the integral from 0 to 2π of 28 dθ?

• 112π

• 84π

• 56π

• 28π