Triple Integrals and Tetrahedrons

The video tutorial explains how to evaluate a triple integral over a solid tetrahedron defined by specific vertices. It covers finding the equation of the plane using intercepts, determining the limits of integration, and setting up the triple integral with respect to different variables. The tutorial also discusses solving for y in terms of x and z, finding the xz trace, and setting integration limits for x and z. The video concludes with a summary and a preview of the next steps in evaluating the integral.

What is the shape of the region over which the triple integral is evaluated?

• Cube

• Sphere

• Cylinder

• Tetrahedron

Which planes bound the region of integration?

• x = 1, y = 1, z = 1

• x = 3, y = 3, z = 3

• x = 0, y = 0, z = 0

• x = 2, y = 2, z = 2

What is the equation of the plane that bounds the tetrahedron?

• 6x + 10y + 15z = 30

• 5x + 3y + 2z = 1

• x + y + z = 1

• 2x + 5y + 3z = 10

In which order is the integration performed in this problem?

• d z, d y, d x

• d y, d x, d z

• d z, d x, d y

• d x, d y, d z

What is the upper limit of integration for y?

• 2x + 5z

• 3 - 3/5 x - 3/2 z

• 5 - 5/2 z

• 10 - 5z

What is the equation used to find the xz trace?

• 2x + 5z = 10

• 3x + 4z = 12

• 6x + 15z = 30

• x + z = 5

What is the x-intercept of the line in the xz trace?

• 6

• 5

• 4

• 3

What is the z-intercept of the line in the xz trace?

• 3

• 1

• 4

• 2

What is the lower limit of integration for z?

• z = 1

• z = 2

• z = 0

• z = 3

What additional traces are mentioned for different orders of integration?

• xy and yz traces

• xz and yz traces

• yz and zx traces

• xy and xz traces