Understanding Double Integrals in Polar Coordinates

The video tutorial explains how to evaluate a double integral over a specified region using polar coordinates. It begins by introducing the integral and the region defined in polar coordinates. The tutorial then demonstrates converting the integral to polar coordinates, setting up the function, and performing integration with respect to r. Finally, it covers substitution and integration with respect to theta, providing both exact and approximate results.

What is the region R defined by in polar coordinates?

• r between 2 and 5, theta between pi and 2pi

• r between 1 and 4, theta between pi/2 and pi

• r between 0 and 3, theta between 0 and pi

• r between 3 and 6, theta between 3/4 pi and 3/2 pi

What is the expression for differential area 'dA' in polar coordinates?

• r dtheta

• r^2 dr dtheta

• dr dtheta

• r dr dtheta

How is the function 2x expressed in terms of r and theta?

• r sin(theta)

• r^2 cos(theta)

• 2r cos(theta)

• 2r sin(theta)

What are the limits of integration for r?

• 0 to 3

• 3 to 6

• 1 to 4

• 2 to 5

What is the integrand function after rewriting in polar coordinates?

• 2r^2 cos(theta)

• r^2 cos(theta)

• 2r^2 sin(theta)

• r^3 cos(theta)

What is the result of integrating 2r^2 cos(theta) with respect to r?

• 2/3 r^3 sin(theta)

• 2/3 r^3 cos(theta)

• r^3 cos(theta)

• r^3 sin(theta)

What is the value of the difference 2/3 times the cube of 6 minus 2/3 times the cube of 3?

• 54

• 126

• 216

• 72

What is the integral of 126 cos(theta) with respect to theta?

• 63 sin(theta)

• 63 cos(theta)

• 126 cos(theta)

• 126 sin(theta)

What is the sine of 3/2 pi?

• sqrt(2)/2

• -1

• 1

• 0

What is the approximate decimal value of the final result rounded to four decimal places?

• -215.0000

• -126.9999

• -126.0000

• -215.9999