Polar Coordinates and Double Integrals

The video tutorial explains how to evaluate a double integral over a semicircular region D, defined by the inequalities x^2 + y^2 ≤ 1 and x ≥ 0. The process involves converting the integral from rectangular to polar coordinates, graphically representing the region and function, setting the limits of integration for R and Theta, and finally evaluating the integral step-by-step. The tutorial emphasizes that the integral does not represent volume in this case and concludes with the final result of 8/3.

What is the shape of the region D over which the double integral is evaluated?

• A full circle

• A semicircle

• A triangle

• A square

Which inequality describes the region D in the XY plane?

• x^2 + y^2 >= 1 and x <= 0

• x^2 + y^2 <= 1 and x >= 0

• x^2 + y^2 < 1

• x^2 + y^2 > 1

What substitution is made for x in the conversion to polar coordinates?

• Theta

• R cos Theta

• R sin Theta

• R

In the graphical representation, what does the double integral not represent?

• Width

• Volume

• Area

• Length

What is the range of the radius R for the region of integration?

• 0 to 2

• 0 to 1

• 0 to 3

• 1 to 2

What are the limits of integration for Theta in the polar form?

• 0 to 2Pi

• -Pi/2 to Pi/2

• 0 to Pi

• -Pi to Pi

What is the common factor in the integrand function that is factored out?

• Theta

• R

• Pi

• Cosine

What is the result of integrating with respect to R?

• R squared

• R to the power of 4

• R cubed divided by 3

• R divided by 2

What is the final value of the double integral?

• 4/3

• 1/3

• 8/3

• 2/3

What is the antiderivative of 4 cosine Theta with respect to Theta?

• 4 cosine Theta

• 4 tangent Theta

• 4 sine Theta

• 4 secant Theta