Surface Area of Revolution Problems

This video tutorial explains how to find the surface area of revolution for parametric functions. It covers the process of calculating the surface area when the function is rotated about the x-axis and y-axis. The tutorial includes detailed steps on using integration and u-substitution techniques to solve the problem, with examples for each axis. The video also provides a comparison of the formulas used for rotations about different axes.

What is the first step in finding the surface area of revolution for a parametric function?

• Find the derivatives dx/dt and dy/dt.

• Directly apply the surface area formula.

• Determine the limits of integration.

• Calculate the volume of revolution.

Which formula is used to find the surface area of revolution about the x-axis?

• 2πx times the integral of the square root of (dx/dt)^2 + (dy/dt)^2

• 2π times the integral of (dx/dt)^2 + (dy/dt)^2

• 2πy times the integral of the square root of (dx/dt)^2 + (dy/dt)^2

• π times the integral of the square root of (dx/dt)^2 + (dy/dt)^2

What substitution is used to simplify the integral in the x-axis rotation problem?

• u = t^3 + 8

• u = t^2 + 4

• u = 4t^2 + 64

• u = 2t + 8

What is the result of evaluating the integral from 64 to 80 in the x-axis rotation problem?

• 203.542

• 715.542

• 512

• 852.459

How does the formula for surface area change when rotating about the y-axis?

• It uses π instead of 2π.

• It uses 2πy instead of 2πx.

• It uses 2πx instead of 2πy.

• It remains the same.

What is the expression for dx/dt in the y-axis rotation problem?

• t^3

• 2t

• t^2

• 3t^2

What substitution is used in the y-axis rotation problem to simplify the integral?

• u = t^3 + 9

• u = 9t^2 + 64

• u = t^2 + 4

• u = 4 + 9t^2

What are the new limits of integration after substitution in the y-axis rotation problem?

• 1 to 4

• 64 to 80

• 4 to 13

• 0 to 1

What is the final expression for the surface area when rotated about the y-axis?

• 2π times 247√13 + 64 over 81

• π times 247√13 + 64 over 1215

• π times 247√13 + 64 over 81

• 2π times 247√13 + 64 over 1215

What is the approximate surface area when rotated about the y-axis?

• 4.9364

• 852.459

• 203.542

• 715.542