Find the volume of doorstop

Question

With the use of a 3d printer, doorstops are being made. The doorstop is a solid whose base is bounded by y = x + 1 and y = x^2 -1, and is formed through cross sections perpendicular to the x-axis are rectangles of height 5 cm. Find its volume

Answer 1

The curves intersect at (1,0) and (2, 3).

The length of a typical rectangular cross section is: [(x + 1) - (x² - 1)] = -x² + x - 2

That means A(x), the area of a typical cross section is 5[-x² + x + 2].

So the volume of the solid with a base as a region between y = x + 1 and y = x² - 1 is:

∫_-1² A(x) dx

5 ∫_-1² (-x² + x + 2)dx

5[-x³/3 + x²/2 + 2x |_-1² ]

5[(-8/3 + 2 + 4) - (1/3 + 1/2 - 2)]

5[ -16/6 + 36/6 - 2/6 - 3/6 + 12/6]

5(27/6)

5(9/2)

45/2 cubic cm

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