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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:

a) 577/15

b) 512/15

c) 572/15

d) 497/15

e) 542/15

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.

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The volume of an element of thickness $\delta x$ is $$(4-x^2)^2\delta x$$

Integrate this with the limits you have and the final answer is b).

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