Consider the functions f(x) = (x^3 / (x^4+1)) and g(x) = (x / (x^4+1)). Let R denote the region in the first quadrant bounded by the curves y = f(x) and y = g(x). Find the exact volume of the solid that has R as its base if every cross section by a plane perpendicular to the x-axis is a rectangle of height 3.

Question

Consider the functions f(x) = (x^3 / (x^4+1)) and g(x) = (x / (x^4+1)). Let R denote the region in the first quadrant bounded by the curves y = f(x) and y = g(x). Find the exact volume of the solid that has R as its base if every cross section by a plane perpendicular to the x-axis is a rectangle of height 3.
I saw someone else ask this question but there were no replies so I wanted to ask it again! Thanks.

oobleck · Answer

So. the curves intersect at (0,0) and (1,1)
Each rectangle has a side of length g(x)-f(x) = (x-x^3)/(x^4+1)
so the volume is just the sum of all those rectangles of thickness dx.
v = ∫[0,1] 3(x-x^3)/(x^4+1) dx = 3/8 (π-ln4)
In case you are having trouble doing that integral, the
x^3/(x^4+1) part is easy, since if u = x^4+1, that is just 1/4 du/u
The tricky one is x/(x^4+1)
If u = x^2, then that is just 1/2 du/(u^2+1) = 1/2 arctan(u)