Triangle Geometry and Integration Concepts

The video tutorial explains how to calculate the total electric charge distributed over a triangular region in the xy-plane. The charge density is given by the function sigma(x, y) = 2xy. The process involves setting up a double integral over the defined region, determining the limits of integration, and evaluating the integral step-by-step. The tutorial concludes with the calculation of the total charge, which is approximately 26.6667 coulombs.

What is the charge density function given in the problem?

• sigma(x, y) = x + y

• sigma(x, y) = 2xy

• sigma(x, y) = x - y

• sigma(x, y) = x^2 + y^2

Which points form the vertices of the triangular region?

• (0, 0), (2, 0), (2, 4)

• (2, 0), (2, 4), (0, 4)

• (0, 2), (2, 2), (2, 4)

• (0, 0), (4, 0), (4, 2)

What is the equation of the line that forms the hypotenuse of the triangle?

• y = 2x - 4

• y = -2x + 4

• y = 4x - 2

• y = 2x + 4

What is the slope of the line forming the hypotenuse of the triangle?

• 1

• -1

• -2

• 2

What are the limits of integration for y in the double integral?

• 0 to 2

• 4 - 2x to 4

• 4 to 0

• 0 to 4

What is the upper limit of integration for x?

• 1

• 4

• 2

• 0

What is the first step in evaluating the double integral?

• Integrate with respect to x

• Integrate with respect to y

• Substitute the limits for y

• Substitute the limits for x

What is the simplified form of the expression after integrating with respect to y?

• x^2 - 4x^3

• x^3 - 4x^2

• 16x^2 - 4x^3

• 16x - 4x^2

What is the final result of the integration with respect to x?

• 80/3

• 26.6667

• 40/3

• 53.3333

What is the total charge over the region in coulombs?

• 40/3

• 26.6667

• 80/3

• 53.3333