Integration Techniques and Derivatives

This video tutorial continues the evaluation of a triple integral over a solid tetrahedron. It begins by setting up the problem and determining the order of integration. The tutorial then walks through the integration process with respect to Y, X, and finally Z, while handling complex substitutions and simplifications. The video concludes with the final result of the integral, emphasizing the algebraic steps involved.

What is the order of integration determined for the triple integral?

• Dx Dz Dy

• Dx Dy Dz

• Dy Dx Dz

• Dz Dy Dx

When integrating with respect to Y, what is treated as a constant?

• X

• Y

• Z

• None

What is the expression for the new integrand function after substituting for Y?

• 3X - 35YZ - 3Y^2

• 3Z - 35XZ - 3Z^2

• 3Z - 35XY - 3Y^2

• 3Y - 35XZ - 3X^2

What substitution is made for X during the integration with respect to X?

• 5 - 5X

• 5 - 5Z

• 5 - 5W

• 5 - 5Y

What is the result of squaring the expression (5 - 5Z)?

• 25 - 50Z + 50Z^2

• 25 - 50Z + 25Z^2

• 25 - 25Z + 50Z^2

• 25 - 25Z + 25Z^2

What is the simplified expression for the Z cubed terms after combining like terms?

• 158

• 158Z

• 158Z^2

• 158Z^3

What is the final step in evaluating the triple integral?

• Integrate with respect to X

• Integrate with respect to Y

• Integrate with respect to Z

• Combine like terms

What is the simplified result of the triple integral evaluation?

• 5

• 1.5

• 2.5

• 10

What is the expression for the N head derivative with respect to Z?

• 158Z^3/3 - 15Z^2/2 + 15Z/1

• 158Z^2/2 - 15Z/1 + 15

• 158Z/1 - 15 + 15Z^2/2

• 158Z^4/4 - 15Z^3/3 + 15Z^2/2

What happens to each term when Z is zero during the final integration?

• Each term becomes zero

• Each term doubles

• Each term halves

• Each term remains unchanged