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 x4+1 x4+1 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. ("Exact volume" means no calculator numbers.)

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 x4+1 x4+1 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. ("Exact volume" means no calculator numbers.)