A rotation over a horizontal line is taken in regards to x's, while a rotation over a vertical line is taken in regards to y's.
2. The difference between creating a washer and a disk is that a washer involves a gap in its rotation. This can be due to having 2 equations, or being bounded by more than one line.
3. Find the volume of the region bounded by y=x, x=3, and x=5, rotated about the x-axis.
Begin by creating a visual representation. Which I do not have.
Then, set up the equation. The interval is [3,5] because of the shaded region, and it is taken in regards to x because it is rotated about a horizontal line. The equation y=x is the radius.
3b. Find the volume of the region bounded by y= x^3, x=1, and x=2, and y=2, rotated about the y-axis.
The equation is on [1,8] because the equation must be taken in regard to y's, which are vertical. The area that won't be there, the middle of the washer, has a radius of 2, so it is subtracted from the 3D radius of the cubed root of y and solved.
AND a throwback to the last time Illinois was decent enough to make the tournament.
