I have been asked this question by many students and I cannot produce the correct solution for them. So here is the original question:
The base of a solid is the region by y=x^2+1 and y=5. Find the volume of the solid given that the cross sections perpendicular to the x-asis are squares.
The answer choices are as follows:
I also know that choice (a) is incorrect.
My attempt at solving the problem was to first find the area of the region, which is the integral from x=-2 to x=+2 of 5-(x^2+1). Now since we are taking cross sections perpendicular to the x-axis that means that the cross sections should be vertical lines such as x=0, x=1 and so on. If one where to take a cross section at a certain x value one could always adjust the height of the solid such that the cross section would be a square. However I don't believe that all of the cross sections would be squares for any fixed height of the object. I found out that the height of the object needs to be less than h=4, but I don't know how to find such h.
So my method was to get the area by integration and then multiply the answer by h (height) to get the volume, but something is incorrect about my logic, because I cannot produce the correct answer. Either I am incorrect or the problem is illogical.