Disk/Washer Method Volume Example Rotating around a Horizontal Line

TL;DR

Find the volume of a solid obtained by rotating a region bounded by three equations about a given axis.

Transcript

and this problem we have a region bounded by the graphs of y equals x y equals 3 and x equals 0 and we have to rotate it about the line y equals 7 and then find the volume of the resulting solid so we're gonna use something called the washer method so we'll start by graphing our region so this is the y axis this will be the x axis so we have x and ... Read More

Key Insights

• ❣️ The region is bounded by the equations y=x, y=3, and x=0.
• ❓ The axis of revolution in this problem is y=7.
• 😃 The big R is the full distance from the far end of the rectangle to the axis of revolution.
• 😝 The little R is the distance from the close end to the axis of revolution.
• 🔇 The volume of the solid can be found using the washer method.
• 😫 The integral to find the volume is set up as π∫(big R^2 - little R^2) dx.
• 🔇 The volume can be calculated by substituting the respective values and solving the integral.

Questions & Answers

Q: What is the axis of revolution in this problem?

The axis of revolution is y=7, which is different from the usual x-axis or y-axis.

Q: How do you determine big R and little R in this problem?

Big R is the full distance from the far end of the rectangle to the axis of revolution (7-x). Little R is the distance from the close end to the axis of revolution (7-3=4).

Q: How do you set up the integral to find the volume?

The integral to find the volume is set up as ∫[0 to 3] of (π[(7-x)^2 - 4^2]) dx.

Q: What is the formula for the volume calculation using the washer method?

The formula for the volume is V = π∫[a to b] (big R^2 - little R^2) dx, where a and b are the limits of integration.

Summary & Key Takeaways

• The problem involves finding the volume of a solid obtained by rotating a region bounded by y=x, y=3, and x=0 about the line y=7.

• The washer method is used to solve the problem.

• The key components of the problem are explained, including big R and little R.