Find the Area of the Surface of Revolution | Summary and Q&A

TL;DR

The video demonstrates how to calculate the area of the surface of revolution when a graph is rotated around the x-axis.

Key Insights

• ❎ The formula for finding the area of the surface of revolution involves integrating the square root of 1 plus the derivative squared of the function being rotated.
• ❓ The distance from the function to the axis of revolution is denoted as R(x) in the formula.
• ❓ Using appropriate substitutions can simplify the integral and make the calculation easier.
• ❓ The definite integral is used to evaluate the area of the surface of revolution.
• ☺️ The final result represents the area obtained when the given graph is rotated around the x-axis.
• ⛔ It is important to carefully evaluate the limits of integration when using substitutions.
• ❓ The constant π is involved in the formula for finding the area of the surface of revolution.

Transcript

in this problem we have to find the area of the surface of revolution that we get when we take this graph and we rotate it about the x-axis so the formula to find the area of the surface of revolution is s equals 2 pi times the definite integral from A to B of R of x times the square root of 1 plus the derivative of F squared and then DX so in all ... Read More

Questions & Answers

Q: What is the formula for finding the area of the surface of revolution?

The formula is s = 2π∫[A,B] R(x)√(1 + (f'(x))^2) dx, where R(x) is the distance from the function to the axis of revolution and f'(x) is the derivative of the function.

Q: How is the formula applied to the given example?

In the example, the function f(x) = 2√x is rotated around the x-axis. The distance to the axis of revolution is the same as the function itself. The derivative of the function is calculated as 1/(2√x), and this is squared to apply it in the formula.

Q: What is the purpose of the substitution used in the integral?

The substitution u = x + 1 is used to simplify the integral. With this substitution, the limits of integration are changed, and the integral becomes easier to evaluate.

Q: What is the final result of the calculation?

The final result of the calculation is 171.258 square units, which represents the area of the surface of revolution obtained by rotating the given graph around the x-axis.

Summary & Key Takeaways

• The video explains the formula for finding the area of the surface of revolution, which involves integrating the square root of 1 plus the derivative squared of the function being rotated.

• The example in the video involves rotating the graph of f(x) = 2√x around the x-axis, and the distance from the function to the axis of revolution is also 2√x.

• The formula is applied step-by-step to find the definite integral, where a substitution is used to simplify the expression, and the final result is calculated to be 171.258.