Volume Calculation Using the Disc Method

The video tutorial explains how to use the disc method to find the volume of a solid formed by rotating a region in the first quadrant bounded by specific curves. It involves solving an equation for Y, graphing the function, and applying the disc method by sketching a rectangle. The tutorial then sets up and evaluates the integral to find the volume, using the power rule to find the antiderivative. Finally, it calculates the exact and approximate volume of the solid.

What method is used to find the volume of the solid obtained by rotating the region in the first quadrant?

• Cavalieri's Principle

• Washer Method

• Disc Method

• Shell Method

Which line is the region rotated about in the given problem?

• x = 1

• x = 0

• y = 0

• y = 1

What is the equation of the curve after solving for y?

• y = x^6

• y = x^(1/6)

• y = x^2

• y = x^(1/2)

What shape is used to approximate the volume of one slice of the solid?

• Circle

• Triangle

• Square

• Rectangle

What is the formula for the volume of a right circular cylinder?

• V = πr^2h

• V = 2πrh

• V = πr^2

• V = 4/3πr^3

What is the expression for the radius R(x) in this problem?

• R(x) = 1 - x^(1/6)

• R(x) = 1 + x^(1/6)

• R(x) = x^(1/6)

• R(x) = x

What is the integral expression for the volume of the solid?

• ∫ from 0 to 1 of (1 - x^(1/6))^2 dx

• ∫ from 0 to 1 of (1 + x^(1/6))^2 dx

• ∫ from 0 to 1 of (x^(1/6))^2 dx

• ∫ from 0 to 1 of (1 - x^(1/3))^2 dx

What is the antiderivative of x^(1/6) used in the integration process?

• x^(1/7) / (1/7)

• x^(7/6) / (7/6)

• x^(1/6) / (1/6)

• x^(6/7) / (6/7)

What is the exact volume of the solid in cubic units?

• π/28

• π/14

• π/7

• π/56

What is the decimal approximation of the volume?

• 0.3366

• 0.4488

• 0.2244

• 0.1122