Volume Calculation Using the Disk Method

The video tutorial explains how to use the disk method to find the volume of a solid generated by rotating a region bounded by y = sine squared x and y = 0 on the interval from 0 to pi about the x-axis. It covers setting up the integral, evaluating it using power-reducing formulas, and finding the anti-derivative through substitution. The tutorial concludes with calculating the exact and approximate volume of the solid.

What is the function that defines the region being rotated to find the volume?

• y = cos^2(x)

• y = sin^2(x)

• y = x^2

• y = tan^2(x)

What is the purpose of sketching a representative rectangle in the disk method?

• To calculate the surface area

• To determine the height of the solid

• To approximate the volume of one slice of the solid

• To find the area of the region

In the disk method, what does R(x) represent?

• The diameter of the disk

• The width of the disk

• The radius of the disk

• The height of the disk

What is the integral expression used to find the volume of the solid?

• ∫ R(x) dx from a to b

• π ∫ R(x)^2 dx from a to b

• ∫ R(x)^2 dx from a to b

• π ∫ R(x) dx from a to b

Why is the power-reducing formula applied in the evaluation of the integral?

• To simplify the integration of an even power of sine

• To convert the sine function into a cosine function

• To simplify the integration of an odd power of sine

• To eliminate the cosine function

What is the result of applying the power-reducing formula to sin^2(x)?

• 1 + cos(2x) / 2

• 1 - cos(2x) / 2

• 1 + sin(2x) / 2

• 1 - sin(2x) / 2

What substitution is used to find the anti-derivative of the term involving cos(2x)?

• u = x

• u = 2x

• u = 4x

• u = x^2

What is the exact volume of the solid after evaluating the integral?

• π^2 / 8

• 3π^2 / 4

• π^2 / 4

• 3π^2 / 8

What is the approximate volume of the solid?

• 5.7011

• 2.7011

• 3.7011

• 4.7011

What is the final step in calculating the volume of the solid?

• Sketching the graph

• Applying the disk method formula

• Evaluating the definite integral

• Finding the derivative