Integration and Trigonometric Values

The video tutorial explains how to evaluate a triple integral over a solid region B, defined by specific intervals for X, Y, and Z. The process involves integrating the function 2Z sinx + y^2 with respect to Z, Y, and X in sequence. The tutorial demonstrates the integration steps, substitution, and simplification to find the exact value of the integral. It also discusses the flexibility of changing the order of integration and provides a detailed explanation of trigonometric function values used in the final evaluation.

What is the closed interval for X in the solid region B?

• From -π/2 to π/2

• From 0 to π

• From -2/3π to 1/3π

• From -π to π

In which order is the integration performed in this problem?

• DX, DZ, DY

• DZ, DY, DX

• DY, DZ, DX

• DX, DY, DZ

What is the result of integrating 2Z sin(x) with respect to Z?

• Z sin(x)

• 2 sin(x)

• 2Z^2 sin(x)

• Z^2 sin(x)

What is the simplified form of the integral after integrating with respect to Z?

• 8 sin(x) + 2y^2

• 9 sin(x) + 3y^2

• 8 sin(x) + y^2

• 9 sin(x) + 2y^2

What is the result of integrating 8 sin(x) with respect to Y?

• 8 y^2 sin(x)

• 8 sin(x) y^2

• 8 y sin(x)

• 8 sin(x) y

What is the simplified result after integrating with respect to Y?

• 32 sin(x) + 112/3

• 16 sin(x) + 112/3

• 32 sin(x) + 128/3

• 16 sin(x) + 128/3

What is the integral of 16 sin(x) with respect to X?

• -16 cos(x)

• 16 cos(x)

• -16 sin(x)

• 16 sin(x)

What is the cosine value of 1/3π?

• 1/2

• -1/2

• √3/2

• -√3/2

What is the cosine value of -2/3π?

• -1/2

• √3/2

• 1/2

• -√3/2

What is the exact value of the given triple integral?

• -16 + 112/3 π

• -16 + 128/3 π

• -16 + 336/9 π

• -16 + 224/9 π