Mass Calculation in Polar Coordinates

The video tutorial explains how to find the mass of a plane region in the first quadrant, bounded by two circles with different radii, using a given density function. The process involves setting up a double integral and converting it to polar coordinates for easier computation. The tutorial walks through the integration steps with respect to both r and theta, using substitution methods to simplify the calculations. The final result is a calculated mass of 68, with a detailed explanation of each step.

What is the density function given in the problem?

• x + y

• x * y

• x^2 + y^2

• x^2 * y^2

What is the radius of the inner circle in the problem?

• 2

• 3

• 4

• 5

Why are polar coordinates used in this problem?

• Because the region is a rectangle

• Because the region is between two circles

• Because the density function is complex

• Because the region is a triangle

What is the expression for x in polar coordinates?

• r^2 * sin(θ)

• r^2 * cos(θ)

• r * cos(θ)

• r * sin(θ)

What are the limits of integration for r?

• 0 to 2

• 2 to 4

• 3 to 5

• 0 to 5

What is the simplified form of the integrand function?

• r^3 * cos(θ) * sin(θ)

• r * cos(θ) * sin(θ)

• r^2 * cos(θ) * sin(θ)

• r^4 * cos(θ) * sin(θ)

What is the result of integrating with respect to r?

• r^2 / 2

• r^4 / 4

• r^3 / 3

• r^5 / 5

What substitution is used for integrating with respect to θ?

• u = sin(θ)

• u = sec(θ)

• u = cos(θ)

• u = tan(θ)

What is the value of sin(π/2)?

• √2/2

• 0

• 1

• 1/2

What is the final calculated mass of the region?

• 102

• 68

• 136

• 34