Integral Evaluation and Antiderivatives

This video tutorial focuses on evaluating triple integrals, starting with setting up the limits of integration and replacing dv with dz, dx, and dy. The instructor demonstrates the process of evaluating each integral step-by-step, including finding antiderivatives and simplifying expressions. The video concludes with a second example to reinforce the concepts, highlighting the similarities between triple and double integrals, with an additional step involved.

What is the first step in setting up a triple integral over a rectangular box?

• Determine the order of integration

• Find the antiderivative of the function

• Evaluate the integral with respect to x

• Simplify the expression

When evaluating the first integral with respect to z, which variables are treated as constants?

• None of the above

• x and z

• z and y

• x and y

What is the antiderivative of z squared when evaluating the first integral?

• z to the fifth over five

• z to the fourth over four

• z squared over two

• z cubed over three

In the second integral evaluation, what is the antiderivative of x cubed?

• x to the fourth over four

• x squared over two

• x to the fifth over five

• x cubed over three

What is the simplified result of the second integral after substituting the limits?

• 4 over 3 y

• 2 over 3 y

• 1 over 3 y

• 3 over 4 y

What is the final result of the third integral evaluation?

• 10 over 3

• 5 over 2

• 8 over 3

• 7 over 3

In the new example, what is the first step in evaluating the integral with respect to z?

• Add a z to the constant

• Subtract a z from the constant

• Divide by a constant

• Multiply by a constant

What is the expression obtained after distributing 4xy in the new example?

• 4xy squared minus 4x squared y

• 4x squared y minus 4xy squared

• 4xy cubed minus 4x squared y

• 4x cubed y minus 4xy cubed

What is the antiderivative of y squared in the new example?

• y squared over two

• y cubed over three

• y to the fifth over five

• y to the fourth over four

What is the final answer for the new example of triple integrals?

• 140 over 5

• 180 over 5

• 162 over 5

• 150 over 5