Cylindrical Shell Method and Substitution

This video tutorial demonstrates how to solve problem 45 from a practice exam using the cylindrical shells method to find the volume of a sphere. The instructor explains the setup of the problem, the application of the cylindrical shells method, and the integration process to arrive at the final volume calculation of the sphere.

What is the main method used in this video to find the volume of the sphere?

• Disk Method

• Pappus's Centroid Theorem

• Cylindrical Shells Method

• Geometric Method

Which equation represents a circle with center at the origin and radius R?

• x^2 + y^2 = R

• x^2 + y^2 = 2R

• x^2 + y^2 = R^2

• x^2 - y^2 = R^2

What fraction of the circle is considered for rotation to form a hemisphere?

• Half

• One-third

• One-fourth

• Full circle

What is the significance of doubling the volume obtained from the hemisphere?

• To adjust for the thickness

• To correct the axis of rotation

• To find the volume of the full sphere

• To account for the missing quadrant

What is the purpose of doubling the volume obtained from the hemisphere?

• To adjust for the thickness

• To correct the axis of rotation

• To find the volume of the full sphere

• To account for the missing quadrant

What is the axis of rotation when using the cylindrical shell method in this problem?

• Origin

• x-axis

• y-axis

• z-axis

What is the initial step in using the cylindrical shell method?

• Calculating the circumference

• Identifying the axis of rotation

• Determining the height

• Finding the radius

What is the role of the cylindrical shell method in this problem?

• To adjust the axis of rotation

• To determine the height

• To calculate the volume

• To find the radius

What is the formula for the circumference of a cylindrical shell?

• πd

• πr^2

• 2πx

• 2πr

What is the height of the cylindrical shell in terms of y?

• y = R

• y = √(R^2 - x^2)

• y = x

• y = R^2