Final answer:
The question is about using integration to calculate the volume of a solid created by revolving a given function around the y-axis. The method of cylindrical shells is used to set up the integral, which is then evaluated to find the volume.
Step-by-step explanation:
The question involves calculating the volume of a solid of revolution, which is a topic in integration and calculus. The volume is found by rotating the function f(x)=x/(x^3+1)^(1/2) around the y-axis from x=1 to x=4. Such a problem can be solved using the method of cylindrical shells, where the volume of each infinitesimally thin shell is V = 2πx ∙ height ∙ thickness. Here, height is given by the function f(x), and the thickness is dx, representing a small change in x. The integral that provides the volume V is:
∫ 2πx ∙ f(x) dx from x=1 to x=4
Substituting the function f(x) into this integral, we get:
∫ 2πx ∙ (x/(x^3+1)^(1/2)) dx from x=1 to x=4
Performing this integration will yield the volume of the solid. Remember to apply proper integration techniques and also use a calculator for integration, as the integral does not have a simple antiderivative.