Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.
$ y = x^2 $ , $ x = y^2 $ ; about $ y = 1 $
Applications of Integration
we are given curves and a line and rest. Find the volume of the solid obtained by rotating this region bounded by these curves. About this line. Were asked to sketch the region the solid and a typical disk or washer. The curves are Y equals X squared. A parabola opening up X. Equals Y squared parabola opening to the right. And the line is the line Y equals one that we're rotating around 1st. I'll sketch the region. You say that. Yeah it would have been really woman. No. Holy shit with rock while you're slicing fucking sword just coming off and then you take your sword breaks. You take away the curved sword. She just now we start off by drawing are Upward facing Parabola has a vertex at the origin and a .11. Then we draw The right word facing Parabola. It has appointed the origin or Vertex and also a .11. And so this region in red. This is our region poison. Now this one point is clear the origin. This other point is the .1, 1. The line that we're going to rotate around is the line y equals one. This line in green. So if we rotate, you know like, well I'm going to mirror this across the line of rotation. Look something a bit like this. Who sometimes we're saying romans one of the big things. Knowledge, Yeah, yeah, What else? Yeah, of the these hills. And first that's why that's why romans build roads is the angus Kononov. So they didn't have little dad, they were good at. Yeah, there were horses. So this is what our solid looks like in blue. And we actually see that this is a has washer cross sections that look like this. His how far west today they were like Austria. Mhm. What? And we see that the washer has an inner radius. I learned. Sure, that's right. With alec baldwin, that's about genghis khan. Well, this is going to be one minus uh mm. What one of them is the only does that mean? No, for real Y equals square root of X. This is one minus root X. And the outer radius. But isn't he is one minus X squared everything. Mhm. It's his G. And therefore using the washer method are volume is equal to excuse me, pi times the integral from Well x ranges from zero up to one of our outer radius which is one minus x squared squared minus the inner radius, which is one minus the square root of x squared dx. This is a pretty straightforward integral, integrating square. It is the same as integrating polynomial. And so this eventually, after several steps should simplify to 11 pi over 30. Mhm. How do you think?