Solid of Revolution (part 4)

TL;DR

Find the volume of the solid figure formed by rotating y = square root of x and y = x squared around the x-axis.

Transcript

Let's continue on with our study of rotation of functions around the x, and we'll soon see the y-axis as well. So let's do a slightly harder example than what we've been doing, but I think it might be obvious how to approach it. So there's my y-axis, there's my x-axis, and in a couple of-- I think it was two problems ago-- we figured out if we had ... Read More

Key Insights

• ☺️ The volume of a solid figure formed by rotating functions around the x-axis can be determined using the disk method.
• 🔇 When there are multiple functions involved, the volume is obtained by subtracting the volume of the inside function from the volume of the outside function.
• 😥 Points of intersection between the functions can be found by setting the equations equal to each other and solving for x.

Questions & Answers

Q: How do you calculate the volume of the solid figure formed by rotating functions around the x-axis?

To calculate the volume, we can use the disk method. Each disk is formed by taking a thin slice perpendicular to the x-axis, with radius equal to the function value and thickness equal to dx. The volume is then found by integrating the areas of these disks.

Q: How do you determine the volume when there are multiple functions involved?

When there are multiple functions, we need to subtract the volume formed by the inside function from the volume formed by the outside function. This is done by subtracting the integral of the inside function from the integral of the outside function.

Q: How do you find the points at which the functions intersect?

To find the points of intersection between two functions, we set the two equations equal to each other and solve for x. In this case, we set y = square root of x equal to y = x squared and solve for x, which gives us x = 0 and x = 1 as the points of intersection.

Q: Why is the volume of the solid figure a hollowed-out cup?

The volume is a hollowed-out cup because the inside function leaves an empty space within the solid figure. When rotating the figure, the inside function creates a void within the volume, resulting in a hollow shape.

Summary & Key Takeaways

• The video discusses finding the volume of the solid figure created by rotating the functions y = square root of x and y = x squared around the x-axis.

• The volume is calculated by subtracting the volume formed by rotating the function y = x squared from the volume formed by rotating the function y = square root of x.

• The final volume is determined to be 3pi/10.