For my calculus class, one of the review questions I'm given is
Find the volume V by rotating the region bounded by $y = 5x-x^2$ and $y = x^2 - 5x + 8$ about the $y$-axis.
I've never learned this concept before in high school or previous mathematics courses, can someone explain how to do this? Would appreciate any help!
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Please see my solution to this exact problem here:
Find the volume of rotation about the y-axis for the region bounded by $y=5x-x^2$, and $x^2-5x+8$
Here is the way to think about it which should help derive the formula. The way to think about this is by splitting the object's volume up into very very thin slices. Suppose these slices are parallel to the x axis, so they cut through at a constant y value. For each thin slice, it's volume is approximately the height of the slice $(dy) $ times the area of the slice when viewed looking down from high above on the y axis. That area is either a ring or a circle, depending on the shape you rotated. It can be found like so: take the radius of the larger circle (formed by the function further out on the x axis at any given y value), and use the formula for area of a circle ($\pi r^2$, where $r=x$ in this case) to find the area of that circle. Do the same process to find the area of the smaller circle which is cut out of the center and subtract its area from the larger one to get that of the ring. Multiply this area by $dy $ and you have the volume of your thin slice. Integrate over all thin slices from the low to upper bound of y to find the volume of your shape! The upper and lower bounds for y should be the y values where your two functions intersect.