The length of the diagonal of the square is , so the length of a side is times , which is . So the area is 2x, not 4x.
- Hollywood
First step?The solid lies between planes perpendicular to the x axis at and . The cross-sections perpendicular to the x axis on the interval are squares whose diagonals run from the parabola to the parabola . Find the volume
is the area of a square
The book says the answer is 16. Something went wrong here.
The square is rotated so that the diagonal is in the x-y plane, so the sides of the square are 45 degrees from horizontal. I imagine it starting as a point (x=0) and growing as it comes toward me (x increasing) until it reaches x=4, where the vertices are (4,2,0), (4,0,2), (4,-2,0), and (4,0,-2).
Is that what you meant by "How could we visualize this?"
- Hollywood
The solid lies between planes perpendicular to the x axis at and . The cross-sections perpendicular to the x axis on the interval are squares whose diagonals run from the parabola to the parabola . Find the volume
Again, answer in book is 16. Where am I going wrong here?
How about this way?
Also wrong
Finally, how about this way?
The solid lies between planes perpendicular to the x axis at and . The cross-sections perpendicular to the x axis on the interval are squares whose diagonals run from the parabola to the parabola . Find the volume
is the area of a square
OK, this way looks better. But still not 16. Maybe the diagonal thing needs to be put in.