Volume of revolution (single integral vs triple integral)

Overview

blackpenredpen finds the volume formed by rotating the region bounded by y = x², y = 9, and x = 0 around the y-axis, first with a single disk-method integral and then with a triple integral. Both methods give 81π/2: the three-dimensional description is x² + z² ≤ y for 0 ≤ y ≤ 9, and polar coordinates in the xz-plane simplify the volume integral.

Key takeaways

For rotation around the y-axis, horizontal slices are disks with radius x = √y, so the volume is π∫₀⁹ y dy = 81π/2.
The rotated solid can be represented in three dimensions by x² + z² ≤ y with 0 ≤ y ≤ 9.
A Cartesian triple integral is possible but uses square-root bounds that couple x and z; polar coordinates in the xz-plane fit the circular cross-sections better.
When changing dx dz to polar coordinates, include the Jacobian factor r; omitting it changes the volume calculation.
The disk method and the triple integral are two descriptions of the same volume, and both yield 81π/2.

Chapters

0:00 Set Up the Rotated Region and Evaluate It with Disks

The bounded region lies between y = x², y = 9, and x = 0; rotating it around the y-axis creates a solid with circular horizontal cross-sections.
At height y, each disk has radius x = √y and thickness dy, so its volume element is π(√y)² dy = πy dy.
Integrating from y = 0 to y = 9 gives π∫₀⁹ y dy = 81π/2.

3:15 Describe the Solid with a Triple Integral

Adding a z-axis turns the rotated solid into a three-dimensional region whose circular cross-sections satisfy x² + z² ≤ y.
The vertical coordinate ranges from y = 0 to y = 9; in Cartesian bounds, x and z depend on one another and on y.
A direct Cartesian setup introduces square-root bounds, such as x between −√(y − z²) and √(y − z²), making the integral awkward to compute.

9:20 Use Polar Coordinates in the xz-Plane to Finish the Volume Integral

Set x = r cos θ and z = r sin θ in the xz-plane, so x² + z² becomes r² and the area element dx dz becomes r dr dθ.
The region becomes 0 ≤ r ≤ √y, 0 ≤ θ ≤ 2π, and 0 ≤ y ≤ 9; the extra factor r is the polar-coordinate Jacobian.
Integrating r from 0 to √y gives y/2; integrating over θ and then y reproduces the disk-method result, 81π/2.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, blackpenredpen.